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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IO. BABPT. BENED.
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        <div xml:id="echoid-div7" type="body" level="1" n="1">
          <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-div730" type="letter" level="4" n="1">
                <pb o="376" rhead="IO. BABPT. BENED." n="388" file="0388" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0388"/>
                <p>
                  <s xml:id="echoid-s4447" xml:space="preserve">Poſſumus etiam probare quod periferia quadrati æqualis triangulo æquilatero
                    <lb/>
                  minor ſit periferia ipſius trianguli æquilateri. </s>
                  <s xml:id="echoid-s4448" xml:space="preserve">Cogita triangulum æquilaterum hic
                    <lb/>
                  ſubſcriptum
                    <var>.d.l.q.</var>
                  cuius baſis
                    <var>.l.q.</var>
                  diuiſa ſit per æqualia à perpendiculari
                    <var>.d.o.</var>
                    <reg norm="deſcri- ptumque" type="context simple">deſcri­
                      <lb/>
                    ptũq́;</reg>
                  ſit rectangulum
                    <var>.o.g.</var>
                  quod æquale erit triangulo
                    <var>.d.l.q.</var>
                  ſed periferia trianguli
                    <lb/>
                  maior eſt periferia rectanguli, nam
                    <var>.l.q.</var>
                  æqualis eſt
                    <var>.o.q.</var>
                  cum
                    <var>.d.g.</var>
                  ſed
                    <var>.q.d.</var>
                  maior eſt
                    <var>.o.
                      <lb/>
                    d.</var>
                  ex .18. primi, vnde
                    <var>.l.d.</var>
                  maior etiam
                    <var>.q.g.</var>
                  cum ex .34. dicti latera oppoſita ipſius re
                    <lb/>
                  ctanguli ſint inuicem æqualia, accipiamus poſtea
                    <var>.e.c.</var>
                  æqualem
                    <var>.o.d.</var>
                  et
                    <var>.c.h.</var>
                  indire-
                    <lb/>
                  ctum æqualem
                    <var>.o.q.</var>
                  circa quem diametrum
                    <var>.e.h.</var>
                  intelligatur circulus
                    <var>.e.i.h.k.</var>
                  et. à pun­
                    <lb/>
                  cto
                    <var>.c.</var>
                  dirigatur perpendicularis
                    <var>.k.i.</var>
                  ad
                    <var>.e.h.</var>
                  vnde ex .3. tertij
                    <var>.c.i.</var>
                  æqualis erit
                    <var>.c.k.</var>
                  & ex
                    <lb/>
                  34. quod fit ex
                    <var>.c.i.</var>
                  in
                    <var>.c.k.</var>
                  hoc eſt quadratum ipſius
                    <var>.c.i.</var>
                  æquale erit ei quod fit .ex
                    <var>.e.c.</var>
                    <lb/>
                  in
                    <var>.c.h.</var>
                  hoc eſt rectangulo
                    <var>.g.o.</var>
                  hoc eſt triangulo
                    <var>.d.l.q.</var>
                  ſed
                    <var>.e.h.</var>
                  eſt dimidium perife-
                    <lb/>
                  rię ipſius rectanguli
                    <var>.g.o.</var>
                  quæ minor eſt di midio periferiæ trianguli
                    <var>.d.l.q.</var>
                  vt vidimus
                    <lb/>
                  et
                    <var>.i.k.</var>
                  eſt dimidium periferię quadrati ipſius
                    <var>.i.c.</var>
                  & minor etiam ipſa
                    <var>.e.h.</var>
                  ex .14. tertij
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4449" xml:space="preserve">quare verum eſt propoſitum.</s>
                </p>
                <figure position="here">
                  <image file="0388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0388-01"/>
                </figure>
                <p>
                  <s xml:id="echoid-s4450" xml:space="preserve">Sed quando periferiæ ſunt inuicem æquales, poſſumus etiam breuiter videre id
                    <lb/>
                  quod ſupradiximus, hoc eſt, quod quadratum, maius ſit triangulo æquilatero. </s>
                  <s xml:id="echoid-s4451" xml:space="preserve">Nam
                    <lb/>
                  cum
                    <var>.b.g.</var>
                  ſeſquitertia ſit ad
                    <var>.b.a.</var>
                  ergo
                    <var>.b.g.</var>
                  erit vt .4. et
                    <var>.b.a.</var>
                  ut .3. vnde
                    <var>.b.q.</var>
                  erit vt .16
                    <lb/>
                  et
                    <var>.b.l.</var>
                  vt .9. et
                    <var>.c.q.</var>
                  vt .8. </s>
                  <s xml:id="echoid-s4452" xml:space="preserve">quare
                    <var>.b.l.</var>
                  maius erit ipſo
                    <reg norm="rectangulo" type="context">rectãgulo</reg>
                    <var>.c.q.</var>
                  ſed
                    <var>.c.q.</var>
                  maius eſt
                    <reg norm="triam" type="context">triã</reg>
                    <lb/>
                  gulo
                    <var>.b.o.g.</var>
                  cum
                    <var>.q.g.</var>
                  quæ æqualis eſt
                    <var>.o.g.</var>
                  maior ſit
                    <var>.o.c.</var>
                  ex .18. vel penultima primi,
                    <lb/>
                  nam ſi
                    <var>.q.g.</var>
                  æqualis eſſet
                    <var>.o.c.</var>
                  </s>
                  <s xml:id="echoid-s4453" xml:space="preserve">tunc
                    <var>.c.q.</var>
                  æqualis eſſet triangulo
                    <var>.b.o.g.</var>
                  ex .41. primi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4454" xml:space="preserve">Alia etiam via maiores noſtri vſi ſunt quæ generalis eſt vt in Theone ſupra Al-
                    <lb/>
                  mageſtum videre eſt, medijs perpendicularibus à centris ad latera figurarum, ſed
                    <lb/>
                  quia
                    <reg norm="differentia" type="context">differẽtia</reg>
                  longitudinum ipſarum perpendicularium alio medio inueniri poteſt,
                    <lb/>
                  eo quo ipſi vſi ſunt, prætermittere nolo quin tibi ſcribam.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4455" xml:space="preserve">Ego enim ita diſcurro.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4456" xml:space="preserve">Sint duæ figuræ iſoperimetrę æquilaterę & æquiangulæ, puta primò trian-
                    <lb/>
                  gulum & quadratum quorum centra ſint
                    <var>.e.</var>
                  et
                    <var>.o.</var>
                  à quibus centris ad latera ſint per-
                    <lb/>
                  pendiculares
                    <var>.e.n.</var>
                  et
                    <var>.o.u.</var>
                  vnde
                    <var>.n.</var>
                  et
                    <var>.u.</var>
                  diuident latera per æqualia vt ſcis, ducantur
                    <lb/>
                  poſtea
                    <var>.e.t.</var>
                  et
                    <var>.o.a.</var>
                  ad angulos dictorum laterum, vnde habebimus angulum
                    <var>.o.a.u.</var>
                    <reg norm="di- midium" type="context">di-
                      <lb/>
                    midiũ</reg>
                  recti, et
                    <var>.e.t.n.</var>
                  tertia pars vnius recti, vt ex te ipſo videre potes, </s>
                  <s xml:id="echoid-s4457" xml:space="preserve">quare angulus </s>
                </p>
              </div>
            </div>
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