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-div730" type="section" level="3" n="41">
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                <p>
                  <s xml:id="echoid-s4457" xml:space="preserve">
                    <pb o="377" rhead="EPISTOLAE." n="389" file="0389" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0389"/>
                  a. ſeſquialter erit angulo
                    <var>.t.</var>
                  quod vt clarius videas cogita lineam
                    <var>.b.d.</var>
                  cuius medietas
                    <lb/>
                  ſit
                    <var>.c.d.</var>
                  tertia verò pars illius ſit
                    <var>.g.d.</var>
                  </s>
                  <s xml:id="echoid-s4458" xml:space="preserve">tunc dico
                    <var>.c.d.</var>
                  ſeſquialteram eſſe ipſi
                    <var>.g.d.</var>
                  ſit enim
                    <lb/>
                    <var>f.d.</var>
                  duplum ipſius
                    <var>.g.d.</var>
                  </s>
                  <s xml:id="echoid-s4459" xml:space="preserve">quare
                    <var>.f.d.</var>
                  erunt duæ tertiæ totius lineę
                    <var>.b.d.</var>
                  & quia eadem pro
                    <lb/>
                  portio eſt totius
                    <var>.b.d.</var>
                  ad
                    <var>.c.d.</var>
                  quæ
                    <var>.f.d.</var>
                  ad
                    <var>.g.d.</var>
                  ergo permutando eadem erit totius
                    <var>.b.
                      <lb/>
                    d.</var>
                  ad
                    <var>.f.d.</var>
                  quæ
                    <var>.c.d.</var>
                  ad
                    <var>.g.d</var>
                  . </s>
                  <s xml:id="echoid-s4460" xml:space="preserve">Sed
                    <var>.b.d.</var>
                  ad
                    <var>.f.d.</var>
                  ſeſquialtera eſt, verum igitur erit quod an-
                    <lb/>
                  gulus
                    <var>.a.</var>
                  ſeſquialter ſit ipſi
                    <var>.t.</var>
                  deinde
                    <var>.t.n.</var>
                  eſt ſeſquitertia ipſi
                    <var>.a.u.</var>
                  vt ſuperius vidimus .
                    <lb/>
                  in eorum duplis. </s>
                  <s xml:id="echoid-s4461" xml:space="preserve">ſcimus etiam
                    <var>.n.e.</var>
                  eſſe dimidium ipſius
                    <var>.t.e.</var>
                  co quod cum
                    <var>.e.t.n.</var>
                  ſit
                    <lb/>
                  tertia pars vnius recti, angulus,
                    <var>t.e.n.</var>
                  erit duo tertia vnius recti, vnde
                    <var>.e.n.</var>
                  erit latus.
                    <lb/>
                  exagoni æquilateris inſcriptibilis circulo cuius diameter ſit
                    <var>.e.t.</var>
                  </s>
                  <s xml:id="echoid-s4462" xml:space="preserve">quare
                    <var>.e.t.</var>
                  dupla erit
                    <lb/>
                  ipſi
                    <var>.e.n.</var>
                  in longitudine, ſed quadrupla in potentia: </s>
                  <s xml:id="echoid-s4463" xml:space="preserve">t.n. vero tripla in potentia ipſi
                    <var>.n.
                      <lb/>
                    e.</var>
                  ex penultima primi, quæ omnia etiam ex .8. tertijdecimi. Eucli. elicere potes, ſed
                    <lb/>
                    <var>c.n.</var>
                  erat ſexquitertia ipſi
                    <var>.a.u.</var>
                  in longitudine, hoc eſt ipſi
                    <var>.o.u.</var>
                  nam
                    <var>.o.u.</var>
                  æqualis eſt ipſi
                    <lb/>
                    <var>a.u</var>
                  . </s>
                  <s xml:id="echoid-s4464" xml:space="preserve">quare
                    <var>.n.t.</var>
                  erit minus quam dupla in potentia ipſi
                    <var>.o.u.</var>
                  hoc eſt, vt .16. ad .9. ergo
                    <lb/>
                  maior proportio erit ipſius
                    <var>.t.n.</var>
                  in potentia ad
                    <var>.n.e.</var>
                  quam ad
                    <var>.o.u.</var>
                  </s>
                  <s xml:id="echoid-s4465" xml:space="preserve">quare etiam in lon
                    <lb/>
                  gitudine, maior proportio erit ipſius
                    <var>.t.n.</var>
                  ad
                    <var>.n.e.</var>
                  quam ad
                    <var>.o.u.</var>
                  vnde
                    <var>.o.u.</var>
                  longior erit
                    <lb/>
                  ipſa
                    <var>.n.e.</var>
                  quod eſt propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4466" xml:space="preserve">Sed ſi
                    <var>.o.a.u.</var>
                  eſſet pentagonus æquilaterus & æquiangulus, ſimiliter probabo per-
                    <lb/>
                  pendicularem
                    <var>.o.u.</var>
                  longiorem eſſe
                    <var>.n.e.</var>
                  ipſius trianguli æquilateri, dummodo ſint iſo-
                    <lb/>
                  perimetrę. </s>
                  <s xml:id="echoid-s4467" xml:space="preserve">Sit enim
                    <var>.a.u.</var>
                  dimidium lateris pentagoni ex ſuppoſito, cuius centrum ſit
                    <lb/>
                  o. </s>
                  <s xml:id="echoid-s4468" xml:space="preserve">tunc proportio
                    <var>.t.n.</var>
                  ad
                    <var>.a.u.</var>
                  erit ſuperbipartienstertias, vt ex ordine iam hic ſupradi
                    <lb/>
                  cto à te facillimè elicere potes, hoc eſt, vt .5. ad .3. et
                    <var>.a.u.</var>
                  minor erit
                    <var>.o.u.</var>
                  eo quod
                    <lb/>
                  angulus
                    <var>.o.</var>
                  minor erit angulo
                    <var>.a.</var>
                  nam angulus
                    <var>.o.</var>
                  erit quinta pars
                    <reg norm="duorum" type="context">duorũ</reg>
                  rectorum, hoc
                    <lb/>
                  eſt duæ quintæ vnius recti, vnde angulus
                    <var>.a.</var>
                  reſiduum vnius recti erit tres quin-
                    <lb/>
                  tæ vnius recti, </s>
                  <s xml:id="echoid-s4469" xml:space="preserve">quare angulus
                    <var>.a.</var>
                  maior ericangulo
                    <var>.o.</var>
                  & conſequenter latus
                    <var>.o.u.</var>
                  ma-
                    <lb/>
                  ius latere
                    <var>.a.u.</var>
                  ſed
                    <var>.t.n.</var>
                  minor eſt quam tripla in potentia ad
                    <var>.a.u.</var>
                  eo quod erit vt .25.
                    <lb/>
                  ad .9. cum in longitudine ſit vt .5. ad .3. ſed dicta
                    <var>.t.n.</var>
                  tripla eſt in potentia ad
                    <var>.e.n.</var>
                  qua-
                    <lb/>
                  re
                    <var>.a.u.</var>
                  maior erit ipſa
                    <var>.e.n.</var>
                  ſed
                    <var>.o.u.</var>
                  maior eſt ipſa
                    <var>.a.u.</var>
                  vt diximus, igitur multo magis
                    <var>.
                      <lb/>
                    o.u.</var>
                  maior eſt ipſa
                    <var>.a.u.</var>
                  vt
                    <reg norm="diximus" type="simple">diximꝰ</reg>
                  &
                    <reg norm="conſequenter" type="context context">cõſequẽter</reg>
                  multo magis
                    <var>.o.u.</var>
                  maior erit ipſa
                    <var>.n.e</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4470" xml:space="preserve">Quotieſcunque enim cognoſcimus proportionem anguli
                    <var>.o.</var>
                  ad angulum
                    <var>.a.</var>
                  quod
                    <lb/>
                  quidem facillimum eſt, nec non proportionem
                    <var>.t.n.</var>
                  ad
                    <var>.a.u.</var>
                  quod, etiam illico cogno-
                    <lb/>
                  ſcitur, </s>
                  <s xml:id="echoid-s4471" xml:space="preserve">tunc exſcientia cordarum & arcuum omnia etiam facillimè innueniuntur.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4472" xml:space="preserve">Verum circa
                    <reg norm="triangulum" type="context">triangulũ</reg>
                  æquilaterum, & pentagonum, alium
                    <reg norm="modum" type="context">modũ</reg>
                  inueni, ſed aliquan
                    <lb/>
                  tulum prolixiorem.</s>
                </p>
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                  <image file="0389-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0389-01"/>
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