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-div630" type="section" level="3" n="26">
              <div xml:id="echoid-div634" type="letter" level="4" n="2">
                <p>
                  <s xml:id="echoid-s3992" xml:space="preserve">
                    <pb o="329" rhead="EPISTOL AE." n="341" file="0341" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0341"/>
                  rem, quæ ita reperientur, efficiemus primo anguium coni, qui ſit
                    <var>.i.A.b.</var>
                  quem diui-
                    <lb/>
                  demus per æqualia mediante
                    <var>.A.q.</var>
                  conſtituendo
                    <var>.A.i.</var>
                  huius anguli æqualem
                    <var>.A.i.</var>
                  ſu-
                    <lb/>
                  perficiei conicæ et
                    <var>.A.q.</var>
                  diuidentem, æqualem parti
                    <var>.A.q.</var>
                  axis coni, ducendo poſtea
                    <lb/>
                  ab
                    <var>.i.</var>
                  per
                    <var>.q.</var>
                  lineam vnam quouſque concurrat
                    <var>.A.b.</var>
                  in puncto
                    <var>.b.</var>
                  habebimus
                    <var>.i.b.</var>
                  pro
                    <lb/>
                  maiori axi ipſi ellipſis, quod per ſe clarum eſt, cuius medietas ſit
                    <var>.i.c.</var>
                  ſed
                    <var>.i.q.</var>
                  ipſius
                    <var>.i.
                      <lb/>
                    b.</var>
                  æqualis eſt ipſi
                    <var>.q.i.</var>
                  ipſius coni, ex quarta primi Eucli. et
                    <var>.q.b.</var>
                  ipſius
                    <var>.i.b.</var>
                  æqualis alte
                    <lb/>
                  ri parti inuiſibili. </s>
                  <s xml:id="echoid-s3993" xml:space="preserve">Reliquum eſt, vt reperiamus minorem axem, quem vocabimus
                    <var>.
                      <lb/>
                    f.r.</var>
                  ducatur ergo primum
                    <var>.q.a.u.n.</var>
                  ad rectos cum
                    <var>.i.b.</var>
                    <reg norm="æqualisque" type="simple">æqualisq́;</reg>
                  ei quæ eſt coni, & diui
                    <lb/>
                  ſa ſimiliter in
                    <var>.a.</var>
                  quæ
                    <var>.u.n.</var>
                  ipſius coni nobis cognita eſt ex lateribus
                    <var>.A.u.</var>
                  et
                    <var>.A.n.</var>
                  & ex
                    <lb/>
                  angulo coni, et
                    <var>.a.q.</var>
                  æqualis eſt
                    <var>.e.p.</var>
                  ex .34. primi. </s>
                  <s xml:id="echoid-s3994" xml:space="preserve">Nunc certi erimus ex .21. primi
                    <lb/>
                  Pergei, quod eadem proportio erit quadrati
                    <var>.u.q.</var>
                  ad quadratum ipſius
                    <var>.f.c.</var>
                  quæ pro-
                    <lb/>
                  ducti ipſius
                    <var>.i.q.</var>
                  in
                    <var>.q.b.</var>
                  ad productum ipſius
                    <var>.i.c.</var>
                  in
                    <var>.c.b.</var>
                  & cum cognita nobis ſint
                    <lb/>
                  hæc tria producta hoc eſt
                    <var>.i.q.</var>
                  in
                    <var>.q.b.</var>
                  et
                    <var>.i.c.</var>
                  in
                    <var>.c.b.</var>
                  et
                    <var>.u.q.</var>
                  in ſeipſa, cognoſcemus
                    <reg norm="etiam" type="context">etiã</reg>
                    <lb/>
                  quartum ipſius
                    <var>.f.c.</var>
                  & fic
                    <var>.f.c.</var>
                    <reg norm="eiuſque" type="simple">eiuſq́;</reg>
                  duplum
                    <var>.f.r.</var>
                  cogniti nobis itaque cum ſint hi duo
                    <lb/>
                  axes
                    <var>.i.b.</var>
                  et
                    <var>.f.r.</var>
                  formabimus ellipſim. </s>
                  <s xml:id="echoid-s3995" xml:space="preserve">Deinde producemus axim
                    <var>.b.i.</var>
                  à part
                    <var>e.i.</var>
                  quo-
                    <lb/>
                  uſque
                    <var>.i.o.</var>
                  æqualis ſit ei quæ extra conum eſt, dein-
                    <lb/>
                    <figure xlink:label="fig-0341-01" xlink:href="fig-0341-01a" number="364">
                      <image file="0341-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0341-01"/>
                    </figure>
                  de ducemus
                    <var>.o.a.</var>
                  quæ circunferentiam ellipticam
                    <lb/>
                  ſecabit in puncto
                    <var>.K.</var>
                  vnde habebimus quantita-
                    <lb/>
                  tem ipſius
                    <var>.o.K.</var>
                  et
                    <var>.K.i.</var>
                  rectam. </s>
                  <s xml:id="echoid-s3996" xml:space="preserve">inde mediante cir-
                    <lb/>
                  cino ſi acceperimus rectam diſtantiam ab
                    <var>.i.</var>
                  ad
                    <var>.K.</var>
                    <lb/>
                  in ellipſi, </s>
                  <s xml:id="echoid-s3997" xml:space="preserve">deinde firmando pedem circini in pun-
                    <lb/>
                  cto
                    <var>.i.</var>
                  in ſuperficie conica, & cum alio ſignando
                    <lb/>
                  lineam vnam curuam ad partem
                    <var>.K.</var>
                  in ſuperficie
                    <lb/>
                  conica, ſumendo poſtea interuallum
                    <var>.o.K.</var>
                  extra el
                    <lb/>
                  lipſim, </s>
                  <s xml:id="echoid-s3998" xml:space="preserve">deinde firmando vnum pedem circini in
                    <lb/>
                  extre mitate gnomonis, cum alio poſtea ſignan-
                    <lb/>
                  do aliam lineam curuam in ſuperficie ipſius coni,
                    <lb/>
                  quæ primam ſe cet in puncto
                    <var>.K.</var>
                  hoc erit punctum
                    <lb/>
                  quæſitum horę propoſitæ in ſuperficie conica
                    <lb/>
                  propoſita.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3999" xml:space="preserve">Sed ſi talis ſectio fuerit parabole, vel hyperbo
                    <lb/>
                  le, tunc mediante ſuo diametro
                    <var>.i.q.</var>
                  cum baſi
                    <var>.u.
                      <lb/>
                    q.n.</var>
                  cognita, deſignabimus ipſam ſectionem
                    <var>.u.i.</var>
                  n
                    <lb/>
                  ope mei
                    <reg norm="inſtrumenti" type="context">inſtrumẽti</reg>
                  in calce meę gnomonicæ de
                    <lb/>
                  ſcripti, </s>
                  <s xml:id="echoid-s4000" xml:space="preserve">deinde diuiſa
                    <var>.u.q.</var>
                  in
                    <var>.a.</var>
                    <reg norm="pro" type="simple">ꝓ</reg>
                    <reg norm="ductaque" type="simple">ductaq́;</reg>
                    <var>q.i.</var>
                    <reg norm="vſque" type="simple">vſq;</reg>
                    <lb/>
                    <figure xlink:label="fig-0341-02" xlink:href="fig-0341-02a" number="365">
                      <image file="0341-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0341-02"/>
                    </figure>
                  ad
                    <var>.o.</var>
                    <reg norm="ductaque" type="simple">ductaq́;</reg>
                    <var>.o.a.</var>
                  habebimus punctum
                    <var>.K</var>
                  . </s>
                  <s xml:id="echoid-s4001" xml:space="preserve">Reli-
                    <lb/>
                  qua facienda ſunt, vt dictum eſt de ellipſi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4002" xml:space="preserve">Inuenta modo cum fuerint duo puncta eiuſ-
                    <lb/>
                  dem horæ propoſitę, ducemus ab vno ad a-
                    <lb/>
                  liud, lineam horariam mediante circino trium
                    <lb/>
                  crurum, quem tibi ſcripſi nudius tertius pro cyl
                    <lb/>
                  lindro, quæ
                    <reg norm="quidem" type="context">quidẽ</reg>
                  linea crit portio gyri ellipſis,
                    <lb/>
                  ſeu hyperbolę, vel parabolę, vt à te ipſo cogi-
                    <lb/>
                  tare potes.</s>
                </p>
              </div>
            </div>
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