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]

Page concordance

< >
Scan Original
401 389
402 390
403 391
404 392
405 393
406 394
407 395
408 396
409 397
410 398
411 399
412 400
413 401
414 402
415 403
416 404
417 405
418 406
419 407
420 408
421 409
422 410
423 411
424 412
425 413
426 414
427 415
428 416
429 417
430 418
< >
page |< < (394) of 445 > >|
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <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-div737" type="section" level="3" n="42">
              <div xml:id="echoid-div737" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s4665" xml:space="preserve">
                    <pb o="394" rhead="IO. BAPT. BENED." n="406" file="0406" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0406"/>
                  iam dicti ad cubum inſius
                    <var>.a.f.</var>
                  ex .11. quinti erit vt dupli
                    <var>.x.n.</var>
                    <reg norm="cum" type="context">cũ</reg>
                  ſimplo
                    <var>.m.n.</var>
                  ad
                    <var>.m.n</var>
                  .</s>
                </p>
                <note position="left" xml:space="preserve">δ</note>
                <p>
                  <s xml:id="echoid-s4666" xml:space="preserve">Superius autem vbi. β. demonſtratum fuit ita eſſe ipſius
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  vt cubi
                    <var>.m.n.</var>
                    <lb/>
                  ad cubum
                    <var>.x.n.</var>
                  & inter. α et. β probatum fuit ita eſſe cubi
                    <var>.a.f.</var>
                  ad cubum
                    <var>.d.g.</var>
                  vt
                    <lb/>
                  cubi
                    <var>.m.n.</var>
                  ad cubum
                    <var>.x.n</var>
                  . </s>
                  <s xml:id="echoid-s4667" xml:space="preserve">Vnde ex .11. quinti
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  erit vt cubi
                    <var>.a.f.</var>
                  ad cubum
                    <lb/>
                    <var>d.g</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4668" xml:space="preserve">Dicit poſtea quod eadem proportio erit inter cubum
                    <var>.d.g.</var>
                  & corpus illud quod
                    <lb/>
                  pro baſi habeat quadratum inſius
                    <var>.d.g.</var>
                  altitudinem verò vt dictum eſt, quæ eſt inter
                    <lb/>
                    <var>d.g.</var>
                  & compoſitum ex duplo
                    <var>.a.f.</var>
                  cum ſimplo
                    <var>.d.g.</var>
                  quod compoſitum eſt altitudo di
                    <lb/>
                  cta, &
                    <reg norm="verum" type="context">verũ</reg>
                  dicit ex ratione ſuperius allegata pro reliquo corpore & cubo ipſius
                    <var>.a.f</var>
                  .
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4669" xml:space="preserve">Quare etiam quemadmodum
                    <var>.t.n.</var>
                  ſe habet ad duplum ipſius
                    <var>.o.n.</var>
                  cum ſimplo
                    <var>.t.n.</var>
                    <lb/>
                  ex ijſdem rationibus ſupradictis, vbiloquuti ſumus de
                    <var>.x.n.</var>
                  cum
                    <var>.m.n</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4670" xml:space="preserve">Diſponantur
                    <reg norm="nunc" type="context">nũc</reg>
                  omnia tali ordine, ita vt
                    <var>.u.</var>
                  primum ſit corpus quod pro ſua ba
                    <lb/>
                  ſi habeat quadratum ipſius
                    <var>.a.f.</var>
                  & c.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4671" xml:space="preserve">Et
                    <var>.y.</var>
                  ſit cubus ipſius
                    <var>.a.f.</var>
                  et
                    <var>.s.</var>
                  ſit cubus ipſius
                    <var>.d.g.</var>
                  et
                    <var>.z.</var>
                  ſit corpus quod baſim ha-
                    <lb/>
                  bet quadratum ipſius
                    <var>.d.g.</var>
                  altitudinem verò vt ſupradictum eſt, et
                    <var>.p.</var>
                  ſit compoſitum
                    <lb/>
                  dupli
                    <var>.n.x.</var>
                  cum ſimplo
                    <var>.m.n.</var>
                  et
                    <var>.l.</var>
                  ſit compoſitum dupli ipſius
                    <var>.n.o.</var>
                  cum ſimplo
                    <var>.t.n.</var>
                    <lb/>
                  Sed
                    <var>.u.</var>
                  locata ſit è regione
                    <var>.p.</var>
                  et
                    <var>.y.</var>
                  è regione
                    <var>.m.n.</var>
                  et
                    <var>.s.</var>
                  è regione
                    <var>.n.t.</var>
                  et
                    <var>.z.</var>
                  è regione
                    <var>.l.</var>
                    <lb/>
                  & habebimus proportionem ipſius
                    <var>.u.</var>
                  ad
                    <var>.y.</var>
                  vt
                    <var>.y.</var>
                  ad
                    <var>.m.n.</var>
                  & ipſius
                    <var>.y.</var>
                  ad
                    <var>.s.</var>
                  vt
                    <var>.m.n.</var>
                  ad
                    <var>.
                      <lb/>
                    n.t.</var>
                  quod ſuperius iam demonſtratum fuit, vbi, δ. et
                    <var>.s.</var>
                  ad
                    <var>.z.</var>
                  ita ſe habebit vt
                    <var>.n.t.</var>
                  ad
                    <var>.
                      <lb/>
                    l.</var>
                  vt vltimò probatum fuit. </s>
                  <s xml:id="echoid-s4672" xml:space="preserve">Quare ex .22. quinti ita ſe habebit
                    <var>.u.</var>
                  ad
                    <var>.z.</var>
                  vt
                    <var>.p.</var>
                  ad
                    <var>.l.</var>
                    <lb/>
                  quemadmodum dicit Archi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4673" xml:space="preserve">Et quia vt ſe habet
                    <var>.u.</var>
                  ad
                    <var>.z.</var>
                  ita facta fuit
                    <var>.h.i.</var>
                  ad
                    <var>.i.K.</var>
                  vbi
                    <var>.R.</var>
                  ideo ex .11. quinti vt ſe
                    <lb/>
                  habet
                    <var>.h.i.</var>
                  ad
                    <var>.i.K.</var>
                  ita ſe habebit
                    <var>.p.</var>
                  ad
                    <var>.l.</var>
                  vt ipſe dicit: </s>
                  <s xml:id="echoid-s4674" xml:space="preserve">Et ex .18. quinti ita erit
                    <var>.h.K.</var>
                    <lb/>
                  ad
                    <var>.K.i.</var>
                  vt
                    <var>.p.l.</var>
                  ad
                    <var>.l.</var>
                  & ex communi conceptu
                    <var>.g.f.</var>
                  ſe habebit ad
                    <var>.h.K.</var>
                  vt quintuplum
                    <lb/>
                  ipſius
                    <var>.p.l.</var>
                  ad
                    <var>.p.l.</var>
                  & ex .22. eiuſdem ita ſe habebit
                    <var>.f.g.</var>
                  ad
                    <var>.i.k.</var>
                  vt quintuplum ipſius
                    <var>.p.
                      <lb/>
                    l.</var>
                  ad
                    <var>.l.</var>
                  quintuplum autem ipſius
                    <var>.p.l.</var>
                  compoſitum eſt ex quintuplo ipſius
                    <var>.n.m.</var>
                  cum
                    <lb/>
                  decuplo ipſius
                    <var>.n.x.</var>
                  cum quintuplo ipſius
                    <var>.n.t.</var>
                  cum decuplo ipſius
                    <var>.n.o.</var>
                  vt à te facilè
                    <lb/>
                  computare potes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4675" xml:space="preserve">Verum etiam erit ex communi ſcientia quod
                    <var>.g.f.</var>
                  ad
                    <var>.f.k.</var>
                  eſt ut quintuplum ipſius
                    <lb/>
                    <var>p.l.</var>
                  ad duplum ipſius
                    <var>.p.l.</var>
                  eo quod ſuperius ſuppoſitum fuit
                    <var>.h.K.</var>
                  eſſe
                    <reg norm="quintam" type="context">quintã</reg>
                  mediam,
                    <lb/>
                  vnde
                    <var>.k.f.</var>
                  relinquebatur pro duabus quintis inferioribus, duplum autem
                    <var>.p.l.</var>
                  com-
                    <lb/>
                  poſitum eſt ex duplo ipſius
                    <var>.m.n.</var>
                  cum duplo ipſius
                    <var>.n.t.</var>
                  cum quadruplo ipſius
                    <var>.n.x.</var>
                  &
                    <lb/>
                  cum quadruplo ipſius
                    <var>.x.o</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4676" xml:space="preserve">Ex conuerſa proportionalitate deinde ita ſe habet,
                    <var>i.K.</var>
                  ad
                    <var>.i.k.</var>
                  ad
                    <var>.f.g.</var>
                  vt
                    <var>.l.</var>
                  ad quin-
                    <lb/>
                  tuplum ipſius
                    <var>.p.l.</var>
                  et
                    <var>.k.f.</var>
                  ad
                    <var>.f.g.</var>
                  vt duplum ipſius
                    <var>.p.l.</var>
                  ad quintuplum ipſius
                    <var>.p.l</var>
                  . </s>
                  <s xml:id="echoid-s4677" xml:space="preserve">Vnde
                    <lb/>
                  ex .24. quinti
                    <var>.i.f.</var>
                  ſe habebit ad
                    <var>.f.g.</var>
                  vt
                    <reg norm="duplum" type="context">duplũ</reg>
                  ipſius
                    <var>.p.l.</var>
                  cum ſimplo
                    <var>.l.</var>
                  ad quintuplum
                    <lb/>
                  ipſius
                    <var>.p.l</var>
                  . </s>
                  <s xml:id="echoid-s4678" xml:space="preserve">Deinde ex conuerſa proportionalitate quintuplum ipſius
                    <var>.p.l.</var>
                  ſe habebit
                    <lb/>
                    <note xlink:label="note-0406-02" xlink:href="note-0406-02a" position="left" xml:space="preserve">θ</note>
                  ad duplum ipſius
                    <var>.p.l.</var>
                  cum ſimplo
                    <var>.l.</var>
                  vt
                    <var>.f.g.</var>
                  ad
                    <var>.f.i</var>
                  . </s>
                  <s xml:id="echoid-s4679" xml:space="preserve">Sed compoſitum dupli ipſius
                    <var>.p.l.</var>
                    <lb/>
                  cum ſimplo
                    <var>.l.</var>
                  æquale eſt duplo ipſius
                    <var>.m.n.</var>
                  cum quadruplo ipſius
                    <var>.x.n.</var>
                  cum ſexcuplo
                    <lb/>
                  ipſius
                    <var>.o.n.</var>
                  cum triplo ipſius
                    <var>.n.t.</var>
                  vt per te computare potes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4680" xml:space="preserve">Superius enim ſumpta fuit
                    <var>.i.r.</var>
                  ad quam ita ſe haberet
                    <var>.f.h.</var>
                  hoc eſt tres quintæ ip-
                    <lb/>
                  ſius
                    <var>.f.g.</var>
                  vt
                    <var>.m.t.</var>
                  ad
                    <var>.t.n</var>
                  . </s>
                  <s xml:id="echoid-s4681" xml:space="preserve">Quare ex conuerſa proportionalitate ita ſe habebit
                    <var>.i.r.</var>
                  ad tres
                    <lb/>
                  quintas ipſius
                    <var>.f.g.</var>
                  vt
                    <var>.t.n.</var>
                  ad
                    <var>.t.m</var>
                  . </s>
                  <s xml:id="echoid-s4682" xml:space="preserve">Et quia
                    <var>.o.n.</var>
                  ſumpta fuit æqualis ipſi
                    <var>.b.g.</var>
                  et
                    <var>.m.n.</var>
                  ipſi
                    <lb/>
                    <var>b.f.</var>
                  ideo
                    <var>.m.o.</var>
                  ex communi ſcientia æ qualis erit ipſi
                    <var>.g.f</var>
                  . </s>
                  <s xml:id="echoid-s4683" xml:space="preserve">Vnde proportio
                    <var>.r.i.</var>
                  ad tres
                    <lb/>
                  quintas ipſius
                    <var>.m.o.</var>
                  erit vt
                    <var>.n.t.</var>
                  ad
                    <var>.t.m.</var>
                  vt inquit Archi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4684" xml:space="preserve">Sed vbi. θ. iam probauimus ita ſe habere
                    <var>.i.f.</var>
                  ad
                    <var>.f.g.</var>
                  vt duplum
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <var>.p.l.</var>
                  cum ſim-
                    <lb/>
                  plo
                    <var>.l.</var>
                  ſe habet ad quintuplum ipſius
                    <var>.p.l.</var>
                  hoc eſt
                    <var>.i.f.</var>
                  ad
                    <var>.m.o.</var>
                  vt duplum ipſius
                    <var>.p.l.</var>
                  cum
                    <lb/>
                  ſimplo
                    <var>.l.</var>
                  ad quintuplum ipſius
                    <var>.p.l</var>
                  .</s>
                </p>
              </div>
            </div>
          </div>
        </div>
      </text>
    </echo>