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-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>
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