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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EPISTOL AE.
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          <div xml:id="echoid-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div670" type="section" level="3" n="29">
              <div xml:id="echoid-div673" type="letter" level="4" n="2">
                <pb o="351" rhead="EPISTOL AE." n="363" file="0363" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0363"/>
                <p>
                  <s xml:id="echoid-s4222" xml:space="preserve">Volo etiam quod ad partem
                    <var>.c.l.s.</var>
                  quadrilateri conſtituta ſit alia parallela ad
                    <var>.z.
                      <lb/>
                    r.</var>
                  & in æquali diſtantia ab ipſa quemadmodum
                    <var>.u.n.</var>
                  diſtat ad eademmet
                    <var>.z.r.</var>
                  ad ean
                    <lb/>
                  dem operationem faciendam. </s>
                  <s xml:id="echoid-s4223" xml:space="preserve">Vnde in vno tantummodo itinere puncti
                    <var>.s.</var>
                  ab
                    <var>.r.</var>
                    <reg norm="vſque" type="simple">vſq;</reg>
                    <lb/>
                  ad
                    <var>.c.</var>
                  deſignabimus quartam partem ſectionis, conuerſo poſtea inſtrumento, hoc eſt
                    <lb/>
                  poſito puncto
                    <var>.r.</var>
                  vbi prius erat
                    <var>.z.</var>
                  et
                    <var>.z.</var>
                  vbi erat
                    <var>.r.</var>
                  aliam delineabimus quartam, &
                    <lb/>
                  ſic ad oppoſitam partem ipſius
                    <var>.z.r.</var>
                  faciendum erit. </s>
                  <s xml:id="echoid-s4224" xml:space="preserve">Hoc inſtrumentum poſſumus
                    <lb/>
                  etiam ita conſtruere, vt puncta
                    <var>.o.</var>
                  et
                    <var>.K.</var>
                  poſſint collocari in laterihus
                    <var>.c.e.</var>
                  et
                    <var>.e.s.</var>
                  vbi no
                    <lb/>
                  bis magis libuerit, ita vt licebit in qualibet proportione
                    <reg norm="axium" type="context">axiũ</reg>
                  propoſita, oxygoniam
                    <lb/>
                  deſignare. </s>
                  <s xml:id="echoid-s4225" xml:space="preserve">Nam
                    <var>.c.o.</var>
                  erit longitudo dimidij axis minoris, et
                    <var>.c.e.</var>
                  dimidij maioris.</s>
                </p>
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            </div>
            <div xml:id="echoid-div676" type="section" level="3" n="30">
              <div xml:id="echoid-div676" type="letter" level="4" n="1">
                <head xml:id="echoid-head515" xml:space="preserve">DE CONSTITVTIONE TRIANGVLI
                  <lb/>
                orthogonij conditionati.</head>
                <head xml:id="echoid-head516" style="it" xml:space="preserve">Domino Ludouico de Rocchaforte.</head>
                <p>
                  <s xml:id="echoid-s4226" xml:space="preserve">
                    <emph style="sc">QVod</emph>
                  à me poſtulas, non eſt admodum difficile, cupis enim triangulum
                    <lb/>
                  orthogonium, exempli gratia
                    <var>.o.i.e.</var>
                  in figura
                    <var>.A.</var>
                  ita conſtituere, vt di-
                    <lb/>
                  uiſum ſit à perpendiculari
                    <var>.a.i.</var>
                  & quod proportio
                    <var>.o.e.</var>
                  ad
                    <var>.o.i.</var>
                  ſit vt
                    <var>.o.i.</var>
                  ad
                    <lb/>
                    <var>i.e.</var>
                  & quod quadrati
                    <var>.o.i.</var>
                  ad quadratum
                    <var>.o.a.</var>
                  ſit vt
                    <var>.e.i.</var>
                  ad
                    <var>.e.a.</var>
                  & quadra
                    <lb/>
                  tum
                    <var>.o.i.</var>
                  ad quadratum
                    <var>.e.i.</var>
                  ſit .ut
                    <var>.o.a.</var>
                  ad
                    <var>.e.a</var>
                  . </s>
                  <s xml:id="echoid-s4227" xml:space="preserve">Quæ omnia in promptu veniunt, quo
                    <lb/>
                  tieſcunque
                    <var>.o.e.</var>
                  fuerit diameter alicuius circuli,
                    <reg norm="diuiſaque" type="simple">diuiſaq́;</reg>
                  in puncto
                    <var>.a.</var>
                  ſecundum pro
                    <lb/>
                  portionem habentem medium
                    <reg norm="duoque" type="simple">duoq́;</reg>
                  extrema, protracta deinde perpendiculari
                    <var>.a.
                      <lb/>
                    i.</var>
                  ad
                    <var>o.e.</var>
                  uſque ad circunferentiam,
                    <reg norm="coniunctæque" type="simple">coniunctæq́;</reg>
                    <var>.o.i.</var>
                  et
                    <var>.i.e</var>
                  : tale triangulum, omnia
                    <lb/>
                  ſupradicta in ſe continebit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4228" xml:space="preserve">Nam ex .30. tertij angulus
                    <var>.i.</var>
                  rectus erit, & ex .8. ſexti
                    <var>.o.i.</var>
                  erit media proportio-
                    <lb/>
                  nalis inter
                    <var>.o.e.</var>
                  et
                    <var>.o.a.</var>
                  et
                    <var>.e.i.</var>
                  inter
                    <var>.o.e.</var>
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0363-01a" xlink:href="fig-0363-01"/>
                  et
                    <var>.a.e.</var>
                  ſed quia ex diuiſione facta in
                    <reg norm="pum" type="context">pũ</reg>
                    <lb/>
                  cto
                    <var>.a.</var>
                  etiam
                    <var>.o.a.</var>
                  erit media proportio-
                    <lb/>
                  nalis inter totum & reſiduum, ideo ex
                    <num value="11">.
                      <lb/>
                    11.</num>
                  quinti ita erit
                    <var>.o.e.</var>
                  ad
                    <var>.e.i.</var>
                  vt
                    <var>.o.e.</var>
                  ad
                    <var>.
                      <lb/>
                    o.a.</var>
                  vnde ex .9. eiuſdem
                    <var>.a.o.</var>
                  erit æqua-
                    <lb/>
                  lis
                    <var>.e.i.</var>
                  & ideo
                    <var>.o.i.</var>
                  erit media proportio
                    <lb/>
                  nalis inter
                    <var>.o.e.</var>
                  et
                    <var>.e.i</var>
                  . </s>
                  <s xml:id="echoid-s4229" xml:space="preserve">Sed quia propor-
                    <lb/>
                  tio
                    <var>.e.i.</var>
                  ad
                    <var>.a.e.</var>
                    <reg norm="eadem" type="context">eadẽ</reg>
                  eſt, quę ipſius
                    <var>.o.e.</var>
                  ad
                    <lb/>
                    <var>o.a</var>
                  . </s>
                  <s xml:id="echoid-s4230" xml:space="preserve">tunc videbis ex .18. ſexti, quod pro
                    <lb/>
                  portio quadrati
                    <var>.o.i.</var>
                  ad quadratum
                    <var>.o.a.</var>
                    <lb/>
                  erit vt
                    <var>.e.i.</var>
                  ad
                    <var>.e.a.</var>
                  cum vero duo trian-
                    <lb/>
                  guli
                    <var>.o.i.a.</var>
                  et
                    <var>.a.i.e.</var>
                  ſint inuicem ſimiles
                    <lb/>
                  ex ſupradicta .8. ſexti, </s>
                  <s xml:id="echoid-s4231" xml:space="preserve">tunc videbis ex
                    <lb/>
                  18. et .17. eiuſdem dictos
                    <reg norm="triangulos" type="context">triãgulos</reg>
                  ean
                    <lb/>
                  dem habere inter ſe proportionem, quę
                    <lb/>
                  eſt inrer quadrata ipſius
                    <var>.o.i.</var>
                  et
                    <var>.i.e.</var>
                  vnde
                    <lb/>
                  ex prima ſexti ita ſe inuicem habebunt
                    <var>.
                      <lb/>
                    a.o.</var>
                  et
                    <var>.a.e</var>
                  .</s>
                </p>
                <div xml:id="echoid-div676" type="float" level="5" n="1">
                  <figure xlink:label="fig-0363-01" xlink:href="fig-0363-01a">
                    <image file="0363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0363-01"/>
                  </figure>
                </div>
                <p>
                  <s xml:id="echoid-s4232" xml:space="preserve">Circa eam verò difficultatem quam </s>
                </p>
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