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-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div586" type="section" level="3" n="21">
              <div xml:id="echoid-div588" type="letter" level="4" n="2">
                <p>
                  <s xml:id="echoid-s3739" xml:space="preserve">
                    <pb o="303" rhead="EPISTOLAE." n="315" file="0315" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0315"/>
                  titudo verò minoris, æqualis ſit ſemidiametro minori, hoc eſt medietati
                    <var>.d.c.</var>
                  vnde
                    <lb/>
                  habebimus proportionem coni maioris ad conum minorem,
                    <reg norm="eandem" type="context">eãdem</reg>
                  quæ eſt diame
                    <lb/>
                  tri maioris ad diametrum minorem, quod ex .2. parte .11. duodecimi Eucli. </s>
                  <s xml:id="echoid-s3740" xml:space="preserve">nec non
                    <lb/>
                  ex .9. eiuſdem manifeſtum eſt, ſed conus minor, eſt quarta pars ſphæroidis prolatæ
                    <lb/>
                  ex .29. </s>
                  <s xml:id="echoid-s3741" xml:space="preserve">Archimedis in lib. de conoidalibus, & conus maior, eſt etiam quarta pars
                    <lb/>
                  ſphæræ, ex .32. primi lib. de ſphæra, & cyllindro, </s>
                  <s xml:id="echoid-s3742" xml:space="preserve">quare ex communi ſcientia,
                    <reg norm="eadem" type="context">eadẽ</reg>
                    <lb/>
                  proportio erit ſphæræ maioris ad ſphæroidem prolatam, quæ
                    <var>.a.b.</var>
                  ad
                    <var>.d.c.</var>
                  ſed pro-
                    <lb/>
                  portio
                    <var>.a.b.</var>
                  ad
                    <var>.d.c.</var>
                  eſt tertia pars proportionis maioris ſphæræ ad
                    <reg norm="minorem" type="context">minorẽ</reg>
                  . </s>
                  <s xml:id="echoid-s3743" xml:space="preserve">Conſidere
                    <lb/>
                  mus
                    <reg norm="nunc" type="context">nũc</reg>
                  alios duos conos rectos, vnius &
                    <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                  baſis,
                    <reg norm="cuius" type="simple">cuiꝰ</reg>
                  diameter ſit
                    <var>.d.c.</var>
                  ſed altitu
                    <lb/>
                  do maioris, æqualis ſit ſemidiametroſphęrę maioris, altitudo verò minoris, ſit æqua
                    <lb/>
                  lis ſemidiametro minoris ſphæræ, vnde ex dictis rationibus habebimus
                    <reg norm="proportio- nem" type="context">proportio-
                      <lb/>
                    nẽ</reg>
                  maioris coni ad
                    <reg norm="minorem" type="context">minorẽ</reg>
                  , vt quæ eſt
                    <var>.o.b.</var>
                  ad
                    <var>.o.d.</var>
                  hoc eſt vt
                    <var>.a.b.</var>
                  ad
                    <var>.d.c.</var>
                  & ex dictis
                    <reg norm="pro­ poſitionibus" type="simple">ꝓ­
                      <lb/>
                    poſitionibus</reg>
                  ita ſe habebit ſphæroides oblonga ad ſphęram minorem vt
                    <var>.a.b.</var>
                  ad
                    <var>.d.
                      <lb/>
                    c.</var>
                  hoc eſt tertia pars proportionis ſphæræ maioris ad minorem. </s>
                  <s xml:id="echoid-s3744" xml:space="preserve">Quare proportio
                    <lb/>
                  ſphæroidis prolatæ ad oblongam, erit reliqua tertia pars proportionis maioris
                    <reg norm="ſphae­ ræ" type="simple">ſphę­
                      <lb/>
                    ræ</reg>
                  ad minorem. </s>
                  <s xml:id="echoid-s3745" xml:space="preserve">Quapropter hæc quatuor corpora continua proportionalia inui-
                    <lb/>
                  cem erunt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3746" xml:space="preserve">Nunc verò quærenda eſt inter
                    <var>.a.b.</var>
                  & ſuas duas tertias partes vna media pro por-
                    <lb/>
                  tionalis, quæ ſit
                    <var>.K.</var>
                  & ex Archimede, inuentum ſit quadratum ęquale circulo, cuius
                    <lb/>
                  ſit
                    <var>.K.</var>
                  diameter. </s>
                  <s xml:id="echoid-s3747" xml:space="preserve">Vnde proportio circuli (cuius
                    <var>.a.b.</var>
                  eſt diameter) ad circulum cu-
                    <lb/>
                  ius
                    <var>.K.</var>
                  eſt diameter, ſeſquialtera erit ex .2. 12. Eucli.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3748" xml:space="preserve">Ducatur deinde quadratum lineæ
                    <var>.K.</var>
                  in lineam
                    <var>.a.b.</var>
                  & proueniet nobis cor-
                    <lb/>
                  pus quoddam, quod æquale erit ſphærę maiori, ex corellario .32. primi de ſphęra &
                    <lb/>
                  cyllindro, cuius corporis, latus cubus ſit
                    <var>.m</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3749" xml:space="preserve">Idem facere oportebit mediante
                    <var>.d.c.</var>
                  minoris ſphærę, cuius corporis cubica ra-
                    <lb/>
                  dix ſit
                    <var>.n</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3750" xml:space="preserve">Nunc verò inter
                    <var>.m.</var>
                  et
                    <var>.n.</var>
                  inueniantur duę medię proportionales
                    <var>.s.t.</var>
                  & ex
                    <var>.s.</var>
                  pro-
                    <lb/>
                  ducatur cubus, qui ęqualis erit ſphęroidi prolatæ propoſiti, cubus vero
                    <var>.t.</var>
                  æqualis
                    <lb/>
                  erit ſphęroidi oblongę, cuius axis eſſet
                    <var>.a.b</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3751" xml:space="preserve">Si autem ſphęroides oblonga nobis propoſita fuiſſet, eodem methodo ſoluere-
                    <lb/>
                  tur problema.</s>
                </p>
              </div>
              <div xml:id="echoid-div589" type="letter" level="4" n="3">
                <head xml:id="echoid-head455" style="it" xml:space="preserve">Quadratum circulis mediantibus deſignare.</head>
                <head xml:id="echoid-head456" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3752" xml:space="preserve">MOdus autem conficiendi quadratum ex circulis ſupra datam lineam, vt Do-
                    <lb/>
                  minum Gaſparem docui, facillimus eſt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3753" xml:space="preserve">Sit enim linea
                    <var>.b.a.</var>
                  46. propoſitionis primi Euclidis,
                    <reg norm="poſitoque" type="simple">poſitoq́;</reg>
                  pede immobli circi-
                    <lb/>
                  ni in puncto
                    <var>.a.</var>
                  ſecundum quantitatem lineæ
                    <var>.a.b.</var>
                  propoſitę fiat circulus, ſimiliter cir-
                    <lb/>
                  ca punctum
                    <var>.b.</var>
                  alius circulus eiuſdem magnitudinis, </s>
                  <s xml:id="echoid-s3754" xml:space="preserve">erecta deinde ſola
                    <var>.a.c.</var>
                  perpendi
                    <lb/>
                  culari ipſi
                    <var>.a.b.</var>
                  ex puncto
                    <var>.a.</var>
                  ipſa ſecabitur à circunferentia circuli. cuius centrum eſt
                    <var>.
                      <lb/>
                    a.</var>
                  in puncto
                    <var>.c.</var>
                  vnde
                    <var>.a.c.</var>
                  æqualis erit
                    <var>.a.b.</var>
                  poſito demum pede immobili ipſius circi
                    <lb/>
                  ni in puncto
                    <var>.c.</var>
                  ſecundum longitudinem ipſius
                    <var>.c.a.</var>
                  fiat alius circulus, qui æqualis erit
                    <lb/>
                  reliquis duobus circulis cum eorum ſemidiametri æquales ſint, & hic vltimo factus
                    <lb/>
                  ſecabit circulum, cuius
                    <reg norm="centrum" type="context">centrũ</reg>
                  eſt
                    <var>.b.</var>
                  in
                    <reg norm="puncto" type="context">pũcto</reg>
                    <var>.d.</var>
                  à quo cum ductæ fuerint
                    <var>.d.c.</var>
                  et
                    <var>.d.b.</var>
                  </s>
                </p>
              </div>
            </div>
          </div>
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