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-div606" type="section" level="3" n="23">
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                <p>
                  <s xml:id="echoid-s3844" xml:space="preserve">
                    <pb o="313" rhead="EPISTOL AE." n="325" file="0325" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0325"/>
                  faciemus, quod diameter
                    <var>.a.b.</var>
                  dictæ ſphæræ ita ſe habcat ad
                    <var>.e.f.</var>
                  ex .10. ſexti, quæ
                    <lb/>
                    <var>e.f.</var>
                  erit reliqua axis quæſita. </s>
                  <s xml:id="echoid-s3845" xml:space="preserve">Vnde conſtituta cum fuerit ellipſis
                    <var>.d.f.t.e.</var>
                  ex dictis axi-
                    <lb/>
                  bus, </s>
                  <s xml:id="echoid-s3846" xml:space="preserve">deinde circumuertendo ellipſim circa maiorem axem, conſtituemus ſphæroi-
                    <lb/>
                  dem oblongam, ſi autem circumuertemus ipſam circa minorem axim conſtituemus
                    <lb/>
                  ſphæroidem prolatam.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3847" xml:space="preserve">Quod autem talis operatio rationalis ſit, nulli dubium erit, que
                    <unsure/>
                  tieſcunque co-
                    <lb/>
                  gnoſcet conum rectum
                    <var>.e.u.f.</var>
                  æqualem eſſe cono recto
                    <var>.a.c.b.</var>
                  ex .2. parte .12. duodeci
                    <lb/>
                  mi Euclid. </s>
                  <s xml:id="echoid-s3848" xml:space="preserve">& quod cum conus
                    <var>.e.d.f.</var>
                  duplus ſit cono
                    <var>.e.u.f.</var>
                  ex lemmate collecto ab
                    <lb/>
                  11. duodecimi, conus
                    <var>.e.d.f.</var>
                  duplus exiſtit etiam cono
                    <var>.a.c.b.</var>
                  ex .7. quinti. </s>
                  <s xml:id="echoid-s3849" xml:space="preserve">Cum de-
                    <lb/>
                  inde ex .32. primi lib. de ſphæra, & cyllindro ſphæra
                    <var>.a.c.b.q.</var>
                  quadrupla ſit cono
                    <var>.a.
                      <lb/>
                    c.b.</var>
                  ipſa conſequenter dupla erit cono
                    <var>.e.d.f.</var>
                  ſed ex .29. primi de conoidalibus, dimi
                    <lb/>
                  dium ſphæroidis
                    <var>.e.d.f.t.</var>
                  hoc eſt
                    <var>.e.d.f.</var>
                  dupla eſt cono
                    <var>.e.d.f</var>
                  . </s>
                  <s xml:id="echoid-s3850" xml:space="preserve">Quare talis medietas
                    <lb/>
                  æqualis eſt ſphæræ propoſitæ, totaq́ue ſphæroides dupla erit ſphærę datæ. </s>
                  <s xml:id="echoid-s3851" xml:space="preserve">Quod
                    <lb/>
                  autem dico de proportione dupla, idem infero de qualibet alia, ſumendo
                    <var>.u.x.</var>
                  ita pro
                    <lb/>
                  portionatam ad
                    <var>.d.x.</var>
                  vt proponitur.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3852" xml:space="preserve">Sphęram autem inuenire quæ dimidia ſit ſphæroidis propoſitæ nullius erit nego-
                    <lb/>
                  tij, quotieſcunque inuentus fuerit modus diuidendi vnam datam proportionem in
                    <lb/>
                  tres æquales partes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3853" xml:space="preserve">Sit propoſita ſphæroides
                    <var>.e.f.d.t.</var>
                  cuius axes ex conſequentia dantur
                    <var>.e.f.</var>
                  et
                    <var>.d.t.</var>
                  quę
                    <lb/>
                  quidem ſphæroides ſit primo oblonga, et
                    <var>.u.x.</var>
                  ſit dimidium axis maioris. </s>
                  <s xml:id="echoid-s3854" xml:space="preserve">imagine-
                    <lb/>
                  tur etiam conus
                    <var>.e.u.f.</var>
                  vt ſupra. </s>
                  <s xml:id="echoid-s3855" xml:space="preserve">Imaginetur etiam factum eſſe, quod proponitur, hoc
                    <lb/>
                  eſt, vt ſphæra
                    <var>.a.b.c.q.</var>
                  ſit dimidium ipſius ſphæroidis, vnde conus
                    <var>.a.c.b.</var>
                  æqualis erit
                    <lb/>
                  cono
                    <var>.e.u.x.</var>
                  vt ſupra demonſtratum eſt, & ſit
                    <var>.g.h.</var>
                  media proportionalis inter
                    <var>.u.x.</var>
                  et
                    <lb/>
                    <var>o.c</var>
                  . </s>
                  <s xml:id="echoid-s3856" xml:space="preserve">Iam viſum ſuperius fuit, quod eadem proportio erat ipſius
                    <var>.u.x.</var>
                  ad
                    <var>.g.h.</var>
                  quæ
                    <var>.a.b.</var>
                    <lb/>
                  ad
                    <var>.e.f.</var>
                  </s>
                  <s xml:id="echoid-s3857" xml:space="preserve">quare eadem quæ
                    <var>.o.b.</var>
                  ad
                    <var>.e.x.</var>
                  ſed
                    <var>.u.x.</var>
                  et
                    <var>.e.x.</var>
                  dantur. </s>
                  <s xml:id="echoid-s3858" xml:space="preserve">inter quas
                    <var>.g.h.</var>
                  et
                    <var>.o.b.</var>
                  vel
                    <lb/>
                    <var>o.c.</var>
                  (nam
                    <var>.o.c.</var>
                  æqualis eſt
                    <var>.o.b.</var>
                  ) medię proportionales ſunt, eo quod cum
                    <var>.g.h.</var>
                  media
                    <lb/>
                  proportionalis ſit inter
                    <var>.u.x.</var>
                  et
                    <var>.o.c.</var>
                  & proportio
                    <var>.o.b.</var>
                  ad
                    <var>.e.x.</var>
                  æqualis ſit ei, quæ
                    <var>.u.x.</var>
                    <lb/>
                  ad
                    <var>.g.h.</var>
                  hoc eſt ei quæ
                    <var>.g.h.</var>
                  ad
                    <var>.o.c.</var>
                  vel. ad
                    <var>.o.b.</var>
                  </s>
                  <s xml:id="echoid-s3859" xml:space="preserve">quare quotieſcunque inuentæ fuerint
                    <var>.
                      <lb/>
                    g.h.</var>
                  et
                    <var>.o.c.</var>
                  vel
                    <var>.o.b.</var>
                  mediæ proportionales inter
                    <var>.d.x.</var>
                  et
                    <var>.x.e.</var>
                  ipſa
                    <var>.o.c.</var>
                  vel
                    <var>.o.b.</var>
                  erit ſemi
                    <lb/>
                  diameter ſphæræ quæſitę. </s>
                  <s xml:id="echoid-s3860" xml:space="preserve">eodem modo faciendum erit ſi ſphęroides fuerit prolata.</s>
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                  <image file="0325-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0325-01"/>
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