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-div518" type="section" level="3" n="8">
              <div xml:id="echoid-div524" type="letter" level="4" n="3">
                <p>
                  <s xml:id="echoid-s3320" xml:space="preserve">
                    <pb o="266" rhead="IO. BABPT. BENED." n="278" file="0278" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0278"/>
                  ſis verò eadem quæ eſt portionis, cuius diameter eſt
                    <var>.n.u.</var>
                  ex .9. 12. Eucli. & ex .42. id-
                    <lb/>
                  eſt vltima primi Archimedis de ſphæra, & cyllindro.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3321" xml:space="preserve">Nunc autem ex hoc aggregato iam vltimo dicto detrahatur conus, cuius
                    <var>.o.a.</var>
                  eſt
                    <lb/>
                  axis et
                    <var>.n.u.</var>
                  diameter baſis, qui quidem conus nobis cognitus eſt, cum
                    <var>.a.n.</var>
                  ſemidia-
                    <lb/>
                  meter eius baſis, nobis cognita ſit ex .34. 3. Eucli. </s>
                  <s xml:id="echoid-s3322" xml:space="preserve">& ſic quantitas eius baſis, & ita ter-
                    <lb/>
                  tia pars
                    <var>.a.o.</var>
                  eius axis, quę multiplicata cum dicta baſi, cuius
                    <var>.n.u.</var>
                  eſt diameter, produ
                    <lb/>
                  cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-
                    <lb/>
                  gre gato, relinquet nobis ſoliditatem portionis
                    <var>.n.e.u.</var>
                  vnde cognoſcemus proportio
                    <lb/>
                  nem iſtius portionis ad totam ſphæram propoſitam.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3323" xml:space="preserve">Sed cum nobis propoſita ſit proportio portionis
                    <var>.n.e.u.</var>
                  ad portionem
                    <var>.i.e.t.</var>
                  cogno
                    <lb/>
                  ſcemus etiam ſoliditatem huius ſecundę portionis
                    <var>.i.e.t.</var>
                  & ſimiliter
                    <reg norm="proportionem" type="context">proportionẽ</reg>
                  hu-
                    <lb/>
                  ius ad totam ſphęram, & ad
                    <reg norm="reſiduum" type="context">reſiduũ</reg>
                    <reg norm="etiam" type="context">etiã</reg>
                  ipſius ſphęrę hoc eſt portioni
                    <var>.i.c.t</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3324" xml:space="preserve">Protrahatur nunc diameter
                    <var>.c.e.</var>
                  à parte
                    <var>.e.</var>
                    <reg norm="vſque" type="simple">vſq;</reg>
                  quo
                    <var>.e.f.</var>
                  æqualis ſit
                    <var>.e.o.</var>
                  ſemidiame
                    <lb/>
                  tro ſphęrę, quæ quidem
                    <var>.f.e.</var>
                  diuidatur in puncto
                    <var>.h.</var>
                  ita vt proportio
                    <var>.f.h.</var>
                  ad
                    <var>.h.e.</var>
                  æqua-
                    <lb/>
                  lis ſit proportioni portionis
                    <var>.i.c.t.</var>
                  ad portionem
                    <var>.i.e.t.</var>
                  quod quidem hoc modo efficie­
                    <lb/>
                  tur. </s>
                  <s xml:id="echoid-s3325" xml:space="preserve">applicabimus lineam
                    <var>.f.q.</var>
                  (indeterminatam) cum
                    <var>.f.e.</var>
                  ad quemuis angulum in
                    <reg norm="pun- cto" type="context">pũ-
                      <lb/>
                    cto</reg>
                    <var>.f.</var>
                  in qua accipiemus duas lineas
                    <var>.f.p.</var>
                  et
                    <var>p.q.</var>
                  inuicem ita relatas, vt ſe habent in pro
                    <lb/>
                  portione duæ iam dictæ portiones, hoc eſt, vt
                    <var>.i.c.t.</var>
                  portio ad portionem
                    <var>.i.e.t.</var>
                  ducen
                    <lb/>
                  do poſtea
                    <var>.q.e.</var>
                  et
                    <var>.p.h.</var>
                  parallelam ad ipſam
                    <var>.q.e.</var>
                  diuiſam habebimus
                    <var>.f.e.</var>
                  in eadem pro
                    <lb/>
                  portione vt dictum eſt ex .2. ſexti, & .11 quinti Euclidis, vnde
                    <var>.c.e</var>
                  :
                    <var>e.f.</var>
                  et
                    <var>.f.h.</var>
                  nobis co
                    <lb/>
                  gnitę erunt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3326" xml:space="preserve">Oportebit nos nunc cognoſcere quantitatem
                    <var>.c.x.</var>
                  hoc modo, videlicet, quęramus
                    <lb/>
                  quadratum, cuius
                    <var>.c.x.</var>
                  eius ſit radix, cui quadratum lineę
                    <var>.c.e.</var>
                  cognitum, ita ſit propor-
                    <lb/>
                  tionatum, vt eſt linea
                    <var>.x.f.</var>
                  ad lineam
                    <var>.f.h.</var>
                  quę nobis cognita eſt, quod rectè factum erit
                    <lb/>
                  ex eo, quod ſcripſit Archimedes in .4. ſecundi de ſphęra, & cyllindro.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3327" xml:space="preserve">Sed quia Archimedes eo in loco ſupponit id, quod necipſe, nec alius adhuc inue
                    <lb/>
                  nit, niſi via naturali, hoc eſt tres partes ęquales ex proportione data effici, non erit in
                    <lb/>
                  conueniens etiam nobis hac via, circa hoc aliquid dicere.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3328" xml:space="preserve">Accipiemus igitur diametrum
                    <var>.c.e.</var>
                  cum addita
                    <var>.e.f.</var>
                  eius ſemidiametro, diuidemus­
                    <lb/>
                  q́ue
                    <var>.f.e.</var>
                  in puncto
                    <var>.h.</var>
                  vt ſupra factum fuit, applicabimus poſtea
                    <var>.c.m.</var>
                  indeterminatam
                    <lb/>
                  angulariter ad
                    <var>.c.e.</var>
                  à qua
                    <var>.c.m.</var>
                  accipiemus
                    <var>.c.g.</var>
                  æqualem
                    <var>.f.h.</var>
                  quęremus deinde natu-
                    <lb/>
                  rali via punctum
                    <var>.b.</var>
                  ita ut protrahendo à puncto
                    <var>.e.</var>
                  (altero extremo diametri)
                    <var>e.m.</var>
                  pa
                    <lb/>
                  rallelam ad
                    <var>.b.g.</var>
                  ductam, erigendo
                    <var>.b.d.</var>
                  perpendicularem ad
                    <var>.c.e.</var>
                  in puncto
                    <var>.b.</var>
                  protra
                    <lb/>
                    <reg norm="ctaque" type="simple">ctaq́;</reg>
                    <var>.d.c.</var>
                  quæ à diametro
                    <var>.e.c.</var>
                  deducta ab
                    <var>.c.</var>
                  incohando vſque ad
                    <var>.x.</var>
                  relinquat nobis
                    <var>.
                      <lb/>
                    x.f.</var>
                  ęqualem
                    <var>.c.m</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3329" xml:space="preserve">Cuius rei ratio eſt, quia quadratum
                    <var>.c.e.</var>
                  ſe habet ad quadratum
                    <var>.c.d.</var>
                  vt
                    <var>.c.e.</var>
                  ad
                    <var>.c.
                      <lb/>
                    b.</var>
                  ex .4. et .18. ſexti Eucl. </s>
                  <s xml:id="echoid-s3330" xml:space="preserve">ſed ex .4. ita ſe habet
                    <var>.m.c.</var>
                  ad
                    <var>.c.g.</var>
                  vt
                    <var>.e.c.</var>
                  ad
                    <var>.b.c.</var>
                  & cum ſit
                    <var>.c.
                      <lb/>
                    g.</var>
                  ęq alis
                    <var>.f.h.</var>
                  ſi
                    <var>.c.m.</var>
                  ęqualis fuerit
                    <var>.f.x.</var>
                  habebimus propoſitum. </s>
                  <s xml:id="echoid-s3331" xml:space="preserve">Quod ſi quis per di-
                    <lb/>
                  ſcretum vel et hoc facere, ita ei agendum erit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3332" xml:space="preserve">Ponamus exempli gratia totum diametrum
                    <var>.c.e.</var>
                  propoſitæ ſphæræ eſſe ut decem,
                    <lb/>
                    <reg norm="proportionemque" type="simple">proportionemq́;</reg>
                  reſiduę portionis
                    <var>.i.c.t.</var>
                  ad ſecundam
                    <var>.i.e.t.</var>
                  hoc eſt
                    <var>.f.h.</var>
                  ad
                    <var>.h.e.</var>
                  ſeſqui-
                    <lb/>
                  alteram eſſe, vnde
                    <var>.e.h.</var>
                  bis tertia erit ìpſius
                    <var>.f.h.</var>
                    <reg norm="totaque" type="simple">totaq́;</reg>
                  linea
                    <var>.c.f.</var>
                  erit .15. et
                    <var>.f.h.</var>
                  erit .3.
                    <lb/>
                  & quadratum lineæ
                    <var>.c.e.</var>
                  erit .100.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3333" xml:space="preserve">Quærendo poſtea quadratum lineæ
                    <var>.c.x.</var>
                  cui quadratum
                    <var>.c.e.</var>
                  hoc eſt .100. ita pro-
                    <lb/>
                  portionatum ſit vt
                    <var>.f.x.</var>
                  ad
                    <var>.f.h.</var>
                  hoc eſt ad .3. ſi autem cogitauerimus
                    <var>.c.x.</var>
                  eſſe nouem
                    <lb/>
                  partium talium qualium
                    <var>.c.e.</var>
                  eſt decem, eius quadratum erit .81. et
                    <var>.x.f.</var>
                  erit .6. par-
                    <lb/>
                  tium talium qualium
                    <var>.c.f.</var>
                  eſt .15. dicendo poſtea ſi .100. dat .81. (ex regula de tribus) </s>
                </p>
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