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
351 339
352 340
353 341
354 342
355 343
356 344
357 345
358 346
359 347
360 348
361 349
362 350
363 351
364 352
365 353
366 354
367 355
368 356
369 357
370 358
371 359
372 360
373 361
374 362
375 363
376 364
377 365
378 366
379 367
380 368
< >
page |< < (341) 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-div642" type="section" level="3" n="28">
              <div xml:id="echoid-div657" type="letter" level="4" n="5">
                <p>
                  <s xml:id="echoid-s4136" xml:space="preserve">
                    <pb o="341" rhead="EPISTOL AE." n="353" file="0353" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0353"/>
                  æquales. </s>
                  <s xml:id="echoid-s4137" xml:space="preserve">Nunc protrahantur duæ
                    <var>.r.o.</var>
                  et
                    <var>.b.o.</var>
                  ab iiſdem punctis
                    <var>.b.r.</var>
                  ad aliud punctum,
                    <lb/>
                  quod volueris ipſius lineæ
                    <var>.p.h.</var>
                  quas probabo
                    <reg norm="longiores" type="conjecture">longiores'</reg>
                  (ſimul ſumptas) eſſe priori-
                    <lb/>
                  bus. </s>
                  <s xml:id="echoid-s4138" xml:space="preserve">Imaginemur igitur duas perpendiculares, ſeu cathetos
                    <var>.b.i.</var>
                  et
                    <var>.q.r.a.</var>
                  punctis
                    <var>.b.
                      <lb/>
                    r.</var>
                  ad
                    <var>.p.h.</var>
                    <reg norm="abſciſſaque" type="simple">abſciſſaq́</reg>
                  ſit linea
                    <var>.o.b.</var>
                  in puncto
                    <var>.x.</var>
                  ita quod
                    <var>.b.x.</var>
                  æqualis ſit ipſi
                    <var>.b.a.</var>
                  quod
                    <lb/>
                  nulli dubium erit poſſe effici, cum
                    <var>.o.b.</var>
                    <reg norm="longiot" type="context">lõgiot</reg>
                  ſit
                    <var>.b.a.</var>
                  co quod opponatur angulo ob-
                    <lb/>
                  tuſo ipſius trianguli
                    <var>.b.a.o.</var>
                  quę
                    <var>.o.b.</var>
                  ſimiliter protrahatur vſque ad
                    <var>.d.</var>
                  ita quod
                    <var>.b.d.</var>
                    <lb/>
                  æqualis ſit
                    <var>.x.b.</var>
                  </s>
                  <s xml:id="echoid-s4139" xml:space="preserve">deinde protrahatur
                    <var>.o.i.</var>
                  quouſque
                    <var>.i.h.</var>
                  æqualis ſit
                    <var>.a.i</var>
                  . </s>
                  <s xml:id="echoid-s4140" xml:space="preserve">In alia parte po-
                    <lb/>
                  ſtea idem faciendum eſt ſecando
                    <var>.a.r.</var>
                  in puncto
                    <var>.u.</var>
                  ita quod
                    <var>.u.r.</var>
                  æqualis ſit
                    <var>.r.o.</var>
                  efficien
                    <lb/>
                  do
                    <var>.r.s.</var>
                  æqualem
                    <var>.r.u.</var>
                  et
                    <var>.q.p.</var>
                  æquale
                    <var>.q.o.</var>
                  vnde habebimus
                    <reg norm="productum" type="context">productũ</reg>
                    <var>.o.d.</var>
                  in
                    <var>.o.x.</var>
                  æqua
                    <lb/>
                  le producto
                    <var>.o.h.</var>
                  in
                    <var>.o.a.</var>
                  & productum
                    <var>.a.s.</var>
                  in
                    <var>.a.u.</var>
                  æquale producto
                    <var>.a.p.</var>
                  in
                    <var>.a.o.</var>
                  exiſtis
                    <lb/>
                  rationibus. </s>
                  <s xml:id="echoid-s4141" xml:space="preserve">Nam cum quadratum ipſius
                    <var>.o.b.</var>
                  æquale ſit duobus quadratis
                    <var>.o.i.</var>
                  et
                    <var>.i.
                      <lb/>
                    b.</var>
                  ex penultima primi Eucli. ipſa quadrata
                    <var>.o.i.</var>
                  et
                    <var>.i.b.</var>
                  æqualia erunt producto
                    <var>.o.d.</var>
                  in
                    <lb/>
                    <var>o.x.</var>
                  ſimul ſumpto cum quadrato
                    <var>.b.x.</var>
                  ex .6. ſecundi, hoc eſt ipſi producto ſimul ſum-
                    <lb/>
                  pto cum quadrato
                    <var>.b.a.</var>
                  hoc eſt ipſi producto ſimul ſumpto cum duobus quadratis
                    <var>.a.
                      <lb/>
                    i.</var>
                  et
                    <var>.i.b.</var>
                  ſed quia productum
                    <var>.o.h.</var>
                  in
                    <var>.o.a.</var>
                  ſimul ſumpto cum quadrato
                    <var>.a.i.</var>
                  ęquatur qua
                    <lb/>
                  drato
                    <var>.o.i.</var>
                  ideo productum
                    <var>.o.h.</var>
                  in
                    <var>.o.a.</var>
                  ſimul ſumptum cum quadrato
                    <var>.a.i.</var>
                  & cum qua-
                    <lb/>
                  drato
                    <var>.i.b.</var>
                  æquale erit producto
                    <var>.o.d.</var>
                  in
                    <var>.o.x.</var>
                  ſimul ſumpto
                    <reg norm="cum" type="context">cũ</reg>
                  duobus quadratis dictis
                    <lb/>
                  hoc eſt ipſius
                    <var>.a.i.</var>
                  et
                    <var>.i.b.</var>
                  quę quadrata dempta cum fuerint ab vtraque parte, tunc cer
                    <lb/>
                  ti erimus producta eſſe inuicem æqualia. </s>
                  <s xml:id="echoid-s4142" xml:space="preserve">Idem dico de alijs ex altera parte. </s>
                  <s xml:id="echoid-s4143" xml:space="preserve">Nunc
                    <lb/>
                  imaginemur protractam eſſc
                    <var>.a.e.</var>
                  parallelam ipſi
                    <var>.o.b.</var>
                  & habebimus proportionem
                    <lb/>
                  ipſius
                    <var>.a.b.</var>
                  ad
                    <var>.a.i.</var>
                  maiorem eſſe ea quæ eſt ipſius
                    <var>.a.e.</var>
                  ad eandem
                    <var>.a.i.</var>
                  cum
                    <var>.a.b.</var>
                  maior
                    <lb/>
                  ſit ipſa
                    <var>.a.e.</var>
                  vt oppoſita angulo obtuſo, quapropter proportio
                    <var>.x.b.</var>
                  ad
                    <var>.a.i.</var>
                  maior erit
                    <lb/>
                  ea quæ eſt
                    <var>.o.b.</var>
                  ad
                    <var>.o.i</var>
                  . </s>
                  <s xml:id="echoid-s4144" xml:space="preserve">Iam enim ſcis proportionem
                    <var>.o.b.</var>
                  ad
                    <var>.o.i.</var>
                  eſſe, vt
                    <var>.a.e.</var>
                  ad
                    <var>.a.i.</var>
                  ex
                    <lb/>
                  ſimilitudine triangulorum. </s>
                  <s xml:id="echoid-s4145" xml:space="preserve">quare proportio
                    <var>.b.d.</var>
                  ad
                    <var>.i.h.</var>
                  maior erit proportione
                    <var>.o.b.</var>
                    <lb/>
                  ad
                    <var>.o.i.</var>
                    <reg norm="tunc" type="context">tũc</reg>
                  ex .27. quinti
                    <reg norm="permutando" type="simple context">ꝑmutãdo</reg>
                    <reg norm="proportio" type="simple">ꝓportio</reg>
                    <var>.b.d.</var>
                  ad
                    <var>.b.o.</var>
                  maior erit proportione
                    <var>.i.h.</var>
                    <lb/>
                  ad
                    <var>.i.o.</var>
                  & ex .26.
                    <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                    <reg norm="componendo" type="context context">cõponẽdo</reg>
                  maior
                    <reg norm="proportio" type="simple">ꝓportio</reg>
                  erit
                    <var>.o.d.</var>
                  ad
                    <var>.o.b.</var>
                  ea quę eſt
                    <var>.o.h.</var>
                  ad. o
                    <lb/>
                  i. &
                    <reg norm="permutando" type="context">permutãdo</reg>
                  maior ipſius
                    <var>.o.d.</var>
                  ad
                    <var>.o.h.</var>
                  ea quæ
                    <var>.o.b.</var>
                  ad
                    <var>.o.i.</var>
                  & ex .33. maior ipſius
                    <var>.b.
                      <lb/>
                    d.</var>
                  ad
                    <var>.i.h.</var>
                  ea quæ
                    <var>.o.d.</var>
                  ad
                    <var>.o.h</var>
                  . </s>
                  <s xml:id="echoid-s4146" xml:space="preserve">Sed vt
                    <var>.b.a.</var>
                  ad
                    <var>.a.i.</var>
                  ita eſt
                    <var>.a.r.</var>
                  ad
                    <var>.a.q.</var>
                  ex ſimilitudine
                    <reg norm="triam" type="context">triã</reg>
                    <lb/>
                  gulorum. </s>
                  <s xml:id="echoid-s4147" xml:space="preserve">Erit igitur
                    <var>.a.r.</var>
                  ad
                    <var>.a.q.</var>
                  maior proportio, ea quæ eſt
                    <var>.o.b.</var>
                  ad
                    <var>.o.i.</var>
                  & exijſdem
                    <lb/>
                  ſupradictis rationibus maior erit proportio ipſius
                    <var>.s.a.</var>
                  ad
                    <var>.p.a.</var>
                  ea quæ eſt
                    <var>.a.r.</var>
                  ad
                    <var>.a.q.</var>
                    <lb/>
                  ſed cum iam probatum fuit proportio
                    <lb/>
                    <figure xlink:label="fig-0353-01" xlink:href="fig-0353-01a" number="384">
                      <image file="0353-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0353-01"/>
                    </figure>
                  nem
                    <var>.b.d.</var>
                  ad
                    <var>.i.h.</var>
                  hoc eſt
                    <var>.a.b.</var>
                  ad
                    <var>.a.i.</var>
                  ma
                    <lb/>
                  iorem eſſe
                    <var>.o.d.</var>
                  ad
                    <var>.o.h.</var>
                  ergo eo ma-
                    <lb/>
                  gis maior erit proportio ipſius
                    <var>.a.s.</var>
                  ad
                    <lb/>
                    <var>a.p.</var>
                  ca quæ
                    <var>.o.d.</var>
                  ad
                    <var>.o.h.</var>
                  ſed cum ex .15
                    <lb/>
                  ſexti, eadem ſit proportio
                    <var>.o.d.</var>
                  ad
                    <var>.o.a.</var>
                    <lb/>
                  quæ
                    <var>.o.h.</var>
                  ad
                    <var>.o.x.</var>
                  et
                    <var>.s.a.</var>
                  ad
                    <var>.o.a.</var>
                  quę
                    <var>a.p.</var>
                    <lb/>
                  ad
                    <var>.a.u.</var>
                  </s>
                  <s xml:id="echoid-s4148" xml:space="preserve">tunc erit
                    <reg norm="permutando" type="context">permutãdo</reg>
                  eadem
                    <lb/>
                  proportio ipſius
                    <var>.o.d.</var>
                  ad
                    <var>.o.h.</var>
                  quæ
                    <var>.o.a.</var>
                    <lb/>
                  ad
                    <var>.o.x.</var>
                  & ipſius
                    <var>.a.o.</var>
                  ad
                    <var>.a.u.</var>
                  quemad-
                    <lb/>
                  modum ipſius
                    <var>.a.s.</var>
                  ad
                    <var>.a.p</var>
                  . </s>
                  <s xml:id="echoid-s4149" xml:space="preserve">Quare maior proportio erit ipſius
                    <var>.a.o.</var>
                  ad
                    <var>.a.u.</var>
                  quam
                    <var>.a.</var>
                  o
                    <unsure/>
                  .
                    <lb/>
                  ad
                    <var>.o.x</var>
                  . </s>
                  <s xml:id="echoid-s4150" xml:space="preserve">Vnde ſequitur
                    <var>.o.x.</var>
                  maiorem eſſe
                    <var>.a.u.</var>
                  ex .8. quinti, ergo
                    <var>.b.x.o.r.</var>
                  longior erit
                    <lb/>
                  ipſa
                    <var>.b.a.u.r</var>
                  . </s>
                  <s xml:id="echoid-s4151" xml:space="preserve">Quod eſt propoſitum.</s>
                </p>
              </div>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum