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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EPISTOLAE.
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          <div xml:id="echoid-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div730" type="section" level="3" n="41">
              <div xml:id="echoid-div733" type="letter" level="4" n="3">
                <p>
                  <s xml:id="echoid-s4480" xml:space="preserve">
                    <pb o="379" rhead="EPISTOLAE." n="391" file="0391" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0391"/>
                  et
                    <var>.b.u.g.</var>
                  ) ſint inuicem æquales ex .4. eiuſdem.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4481" xml:space="preserve">Accipiatur deinde vel intelligatur
                    <var>.g.p.</var>
                  æqualis duabus te
                    <unsure/>
                  rtijs ipſius
                    <var>.a.g.</var>
                  ducatur­
                    <lb/>
                  q́ue
                    <var>.b.p.</var>
                  quam probabo maiorem eſſe duplo ipſius
                    <var>.a.p.</var>
                  vnde maior erit latere ipſius
                    <lb/>
                  trigoni æquilateris, cuius dimidium eſt
                    <var>.a.p.</var>
                  ſcimus enim ipſum latus ſe habere ad
                    <var>.m.
                      <lb/>
                    g.</var>
                  vt quinque ad .3. ita etiam
                    <var>.a.p.</var>
                  ad
                    <var>.a.g.</var>
                  vt diximus.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4482" xml:space="preserve">Cum
                    <reg norm="autem" type="context">autẽ</reg>
                  angulus
                    <var>.a.b.g.</var>
                  ſit quarta pars anguli
                    <var>.b.g.a.</var>
                  ex .10. quarti & quinta pars
                    <lb/>
                  vnius recti ex .32. primi, dictus angulus erit graduum .18. et
                    <var>.a.g.</var>
                  erit partium .30902.
                    <lb/>
                  et
                    <var>.a.b.</var>
                  partium .95015 et
                    <var>.a.p.</var>
                  51503. vnde ex penultima primi latus
                    <var>.b.p.</var>
                  erit par-
                    <lb/>
                  tium .108075. duplum vero ipſius
                    <var>.a.p.</var>
                  erit .103006. latus igitur dicti trigoni, quod
                    <lb/>
                  ab
                    <var>.p.</var>
                  erigitur, ſecabit perpendicularem
                    <var>.a.b.</var>
                  ſub
                    <var>.b.</var>
                  hoc eſt inter
                    <var>.b.</var>
                  et
                    <var>.a.</var>
                  ex penultima
                    <lb/>
                  primi. </s>
                  <s xml:id="echoid-s4483" xml:space="preserve">Finiatur enim triangulus æquicrurus
                    <var>.b.q.p.</var>
                  quem probaui maiorem eſſe æ-
                    <lb/>
                  quilatero iſoperimetro pentagono propoſito,
                    <reg norm="ducaturque" type="simple">ducaturq́;</reg>
                    <var>.u.p.</var>
                  ducatur etiam
                    <var>.u.n.</var>
                  pa-
                    <lb/>
                  rallela ipſi
                    <var>.b.g.</var>
                  quæ concludet triangulum
                    <var>.g.u.n.</var>
                  ſimilem triangulo
                    <var>.m.b.g.</var>
                  eo quod
                    <lb/>
                  cum angulus
                    <var>.m.b.g.</var>
                  æqualis ſit angulo
                    <var>.b.g.u.</var>
                  ex .16. tertij, per .27. primi
                    <var>.m.b.</var>
                  et
                    <var>.g.u.</var>
                    <lb/>
                  erunt inuicem
                    <reg norm="æquidiſtantes" type="context">æquidiſtãtes</reg>
                  , vnde angulus
                    <var>.b.m.g.</var>
                  æqualis erit angulo
                    <var>.u.g.n.</var>
                  et. ex .29.
                    <lb/>
                  angulus
                    <var>.g.u.n.</var>
                  æqualis erit angulo
                    <var>.u.g.b</var>
                  . </s>
                  <s xml:id="echoid-s4484" xml:space="preserve">quare etiam angulo
                    <var>.g.b.m.</var>
                  & angulus
                    <var>.u.n.
                      <lb/>
                    g.</var>
                  angulo
                    <var>.b.g.m.</var>
                  ex .32. eiuſdem, </s>
                  <s xml:id="echoid-s4485" xml:space="preserve">vnde ex .4. ſexti proportio
                    <var>.g.n.</var>
                  ad
                    <var>.g.m.</var>
                  erit .vt
                    <var>.g.u.</var>
                    <lb/>
                  ad
                    <var>.m.b.</var>
                  ſed cum
                    <var>.g.u.</var>
                  maior ſit dimidio ipſius
                    <var>.b.g.</var>
                  ex .20. primi, hoc eſt maior dimi-
                    <lb/>
                  dio ipſius
                    <var>.b.m.</var>
                  ergo
                    <var>.g.n.</var>
                  etiam maior erit ipſa
                    <var>.g.a.</var>
                  quapropter maior erit ipſa
                    <var>.g.p.</var>
                    <lb/>
                  cum
                    <var>.g.p.</var>
                  minor ſit ipſa
                    <var>.g.a.</var>
                  ex hypotheſi, ducta deinde cum fuerit
                    <var>.b.n.</var>
                  habebimus
                    <lb/>
                  triangulum
                    <var>.b.n.g.</var>
                    <reg norm="æqualem" type="context">æqualẽ</reg>
                  triangulo
                    <var>.b.u.g.</var>
                  &
                    <reg norm="maiorem" type="context">maiorẽ</reg>
                    <reg norm="triangulo" type="context">triãgulo</reg>
                    <var>.b.p.g.</var>
                  ex prima ſexti
                    <lb/>
                  vel quia totum maius eſt ſua parte. </s>
                  <s xml:id="echoid-s4486" xml:space="preserve">Triangulus igitur
                    <var>.b.u.g.</var>
                  maior eſt triangu-
                    <lb/>
                  lo
                    <var>.b.p.g</var>
                  . </s>
                  <s xml:id="echoid-s4487" xml:space="preserve">quare triangulus
                    <var>.b.u.o.</var>
                  maior erit triangulo
                    <var>.g.o.p.</var>
                  ex communi conceptu,
                    <lb/>
                  idem infero ab alia parte dictarum figurarum. </s>
                  <s xml:id="echoid-s4488" xml:space="preserve">Quare pentagonus
                    <var>.b.d.m.g.u.</var>
                  maior
                    <lb/>
                  erit triangulo
                    <var>.b.q.p.</var>
                  quem probauimus maiorem eſſe triangulo æquilatero ſibi iſo-
                    <lb/>
                  perimetro.</s>
                </p>
                <figure position="here">
                  <image file="0391-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0391-01"/>
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              </div>
              <div xml:id="echoid-div734" type="letter" level="4" n="4">
                <head xml:id="echoid-head560" style="it" xml:space="preserve">Comparatio periferiarum quadrati & trianguli aquilateri circunſcriptorum ab eodem circulo.</head>
                <head xml:id="echoid-head561" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s4489" xml:space="preserve">QVod autem periferia quadrati in eodem circulo inſcripti, in quo ſit triangu-
                    <lb/>
                  lus æquilaterus, longior ſit periferia ipſius trianguli æquilateri, abſque vllo </s>
                </p>
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