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-div7" type="body" level="1" n="1">
          <div xml:id="echoid-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div680" type="section" level="3" n="31">
              <div xml:id="echoid-div680" type="letter" level="4" n="1">
                <pb o="354" rhead="IO. BABPT. BENED." n="366" file="0366" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0366"/>
                <head xml:id="echoid-head519" style="sc" xml:space="preserve">COROLLARIVM.</head>
                <p>
                  <s xml:id="echoid-s4256" xml:space="preserve">Proportio maioris portionis ad minorem ſemper erit ſeſquialtera proportioni
                    <lb/>
                  ipſius
                    <var>.b.g.</var>
                  ad
                    <var>.a.b.</var>
                  eo quod cum ſit proportio totalis portionis ad partialem vt trian-
                    <lb/>
                  guli
                    <var>.b.g.e.</var>
                  ad
                    <var>.b.a.d.</var>
                  & hæc ſeſquialtera proportioni ipſius
                    <var>.g.e.</var>
                  ad
                    <var>.a.o.</var>
                  hoc eſt vt ip-
                    <lb/>
                  ſius
                    <var>.b.g.</var>
                  ad
                    <var>.b.a.</var>
                  ideo proportio ipſarum portionum erit ſimiliter ſeſquialtera pro-
                    <lb/>
                  portioni diametrorum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4257" xml:space="preserve">Deinde ſi protractæ fuerint
                    <var>.b.d.</var>
                  et
                    <var>.g.e.</var>
                  quouſque conueniant in puncto
                    <var>.z.</var>
                  habe
                    <lb/>
                  bis inter
                    <var>.g.z.</var>
                  et
                    <var>.a.o.</var>
                  duas
                    <var>.g.e.</var>
                  et
                    <var>.a.d.</var>
                  medias proportionales in proportionalitate con
                    <lb/>
                  tinua, eo quod cum (ex ijs quæ ſupra diximus.).
                    <var>a.d.</var>
                  media proportionalis ſit inter
                    <var>.
                      <lb/>
                    g.e.</var>
                  et
                    <var>.a.o.</var>
                  & proportio
                    <var>.g.z.</var>
                  ad
                    <var>.g.e.</var>
                  vt ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.a.o.</var>
                  eo quodipſius
                    <var>.g.z.</var>
                  ad
                    <var>.a.d.</var>
                    <lb/>
                  & ipſius
                    <var>.g.e.</var>
                  ad
                    <var>.a.o.</var>
                  eſt vt ipſius
                    <var>.b.g.</var>
                  ad
                    <var>.b.a.</var>
                  ex ſimilitudine triangulorum, ideo di-
                    <lb/>
                  ctæ
                    <reg norm="proportiones" type="simple">ꝓportiones</reg>
                  erunt
                    <reg norm="inuicem" type="context">inuicẽ</reg>
                  æquales. </s>
                  <s xml:id="echoid-s4258" xml:space="preserve">Vnde permutatim ita erit ipſius
                    <var>.g.z.</var>
                  ad
                    <var>.g.e.</var>
                    <lb/>
                  vt ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.a.o.</var>
                  & ut ipſius
                    <var>.g.e.</var>
                  ad
                    <var>.a.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4259" xml:space="preserve">Amplius etiam dico, quod proportio pa
                    <lb/>
                    <figure xlink:label="fig-0366-01" xlink:href="fig-0366-01a" number="403">
                      <image file="0366-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0366-01"/>
                    </figure>
                  rabolæ totalis ad partialem, eadem eſt, quę
                    <lb/>
                  cubi ipſius
                    <var>.g.e.</var>
                  ad cubum ipſius
                    <var>.a.d.</var>
                  & ex
                    <reg norm="con" type="context">cõ</reg>
                    <lb/>
                  ſequenti, vt cuborum earundem baſium, eo
                    <lb/>
                  quod cum ſit, ex .36. vndecimi Euclid. pro-
                    <lb/>
                  portio cubi ipſius
                    <var>.g.e.</var>
                  ad cubum ipſius
                    <var>.a.d.</var>
                    <lb/>
                  tripla ei quæ ipſius
                    <var>.g.e.</var>
                  ad
                    <var>.a.d.</var>
                  ideo æqualis
                    <lb/>
                  erit ei quę trianguli
                    <var>.b.g.e.</var>
                  ad triangulum
                    <var>.b.
                      <lb/>
                    a.d.</var>
                  cum proportio horum duorum triangu
                    <lb/>
                  lorum compoſita ſit (vt ſupra vidimus) ex
                    <lb/>
                  ea quæ
                    <var>.g.e.</var>
                  ad
                    <var>.a.o.</var>
                  & ex ea quæ
                    <var>.g.e.</var>
                  ad
                    <var>.a.d.</var>
                    <lb/>
                  & hæc medietas illius, ſed trianguli ita ſe in
                    <lb/>
                  uicem habenr, vt parabolę, </s>
                  <s xml:id="echoid-s4260" xml:space="preserve">quare ipſæ para-
                    <lb/>
                  bolæ ſeinuicem habebunt, vt cubi ipſarum
                    <lb/>
                  baſium.</s>
                </p>
              </div>
              <div xml:id="echoid-div683" type="letter" level="4" n="2">
                <head xml:id="echoid-head520" style="it" xml:space="preserve">Cubum fabricare æqualem pyramidi propoſitæ.</head>
                <head xml:id="echoid-head521" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s4261" xml:space="preserve">CVbum fabricare æqualem propoſitæ pyramidi quadrilateræ, nullius erit diffi-
                    <lb/>
                  cultatis, ſuppoſita tamen pro reperta diuiſione cuiuſuis datæ proportionis in
                    <lb/>
                  tres partes æquales. </s>
                  <s xml:id="echoid-s4262" xml:space="preserve">Nam ex .6. duodecimi Eucli. patet omne corpus ſerratile d-ui
                    <lb/>
                  ſibile eſſe in tres pyramides quadrilateras æquales, ſcimus etiam quod cuilibet py-
                    <lb/>
                  ramidi quadrilateræ poteſt reperiri ſuum ſerratile. </s>
                  <s xml:id="echoid-s4263" xml:space="preserve">Sit igitur propoſita pyramis qua
                    <lb/>
                  drilatera
                    <var>.m.g.f.h.</var>
                  cuius ſerratile ita inueniemus, ducendo primum
                    <var>.h.i.</var>
                  parallelam
                    <lb/>
                  ipſi
                    <var>.g.f.</var>
                  et
                    <var>.f.i.</var>
                  ipſi
                    <var>.g.h.</var>
                  in ſuperficie trianguli
                    <var>.f.g.h.</var>
                  et
                    <var>.m.K.</var>
                  ipſi
                    <var>.g.h.</var>
                  in ſuperficie
                    <lb/>
                  trianguli
                    <var>.m.g.h.</var>
                  & æqualem dictæ
                    <var>.g.h.</var>
                  ducetur poſtea
                    <var>.K.h.</var>
                  et
                    <var>.K.i.</var>
                  & habebimus cor
                    <lb/>
                  pus
                    <var>.f.K.g.</var>
                  ſerratile, & triplum pyramidi propoſitæ. </s>
                  <s xml:id="echoid-s4264" xml:space="preserve">Nunc duplicemus ipſum, du-
                    <lb/>
                  cendo
                    <var>.K.x.</var>
                  in ſuperficie trianguli
                    <var>.i.k.h.</var>
                  parallelam,
                    <reg norm="æqualemque" type="simple">æqualemq́;</reg>
                  ipſi
                    <var>.i.h.</var>
                  et
                    <var>.m.y.</var>
                    <lb/>
                  in ſuperficie trianguli
                    <var>.f.m.g.</var>
                  parallelam, ę
                    <reg norm="qualemque" type="simple">qualemq́;</reg>
                  ipſi
                    <var>.f.g.</var>
                  ducatur poſtea
                    <var>.g.y.</var>
                  et
                    <var>.h.
                      <lb/>
                    x.</var>
                  quarum
                    <reg norm="vnaquæque" type="simple">vnaquæq;</reg>
                  æqualis erit ipſi
                    <var>.f.m.</var>
                  vnde habebimus corpus
                    <var>.f.x.</var>
                  parallelepe-
                    <lb/>
                  pidum, & ſexcuplum ipſi pyramidi propoſitæ.</s>
                </p>
              </div>
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