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-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div708" type="section" level="3" n="36">
              <div xml:id="echoid-div708" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s4356" xml:space="preserve">
                    <pb o="365" rhead="EPISTOL AE." n="377" file="0377" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0377"/>
                  ſit angulo
                    <var>.a.h.b.</var>
                  propter æquidiſtantiam dictam, æqualis etiam erit angulo
                    <var>.d.</var>
                  & ar-
                    <lb/>
                  cus
                    <var>.a.x.</var>
                  æqualis arcui
                    <var>.a.g.</var>
                  vnde angulus
                    <var>.a.i.x.</var>
                  æqualis erit
                    <var>.d.</var>
                  ſed angulus
                    <var>.i.a.d.</var>
                  com-
                    <lb/>
                  munis eſt triangulis
                    <var>.c.a.d.</var>
                  et
                    <var>.i.a.t</var>
                  . </s>
                  <s xml:id="echoid-s4357" xml:space="preserve">quare angulus
                    <var>.a.t.i.</var>
                  rectus erit, vt
                    <var>.c.</var>
                  hoc eſt
                    <var>.i.x.</var>
                  per
                    <lb/>
                  pendicularis erit ipſi
                    <var>.a.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4358" xml:space="preserve">Sed vbitibi ſcripſi circa finem illius epiſtolæ, Tartaleam erraſſe in quinta propo-
                    <lb/>
                  ſitione primi lib. ſuæ nouæ ſcientiæ, non ſine ratione illud ſcripſi. </s>
                  <s xml:id="echoid-s4359" xml:space="preserve">Nam, inquit ipſe,
                    <lb/>
                  nullum corpus æquè graue poteſt in aliquo temporis ſpatio moueri motu naturali,
                    <lb/>
                    <reg norm="violentoque" type="simple">violentoq́;</reg>
                  ſimul miſtis. </s>
                  <s xml:id="echoid-s4360" xml:space="preserve">Vbi decipitur, eo quod non animaduertit incrementum ve
                    <lb/>
                  locitatis vnius motus, ſimul eſſe cum decremento velocitatis alterius,
                    <reg norm="eodemque" type="simple">eodemq́;</reg>
                  tem
                    <lb/>
                  pore, vt manifeſtè patet in itinere corporis, ab ipſo pro exemplo aſſumpto, hoc eſt
                    <lb/>
                  quod velocitas motus in ſpatio
                    <var>.c.d.</var>
                  creſcit vt naturalis, & decreſcit vt violenta. </s>
                  <s xml:id="echoid-s4361" xml:space="preserve">
                    <reg norm="nam" type="context">nã</reg>
                    <lb/>
                  creſcit orizontem verſus & decreſcit in remotione à linea
                    <var>.a.b.</var>
                  ſed ſi à puncto
                    <var>.c.</var>
                  ad
                    <lb/>
                  punctum
                    <var>.d.</var>
                  motus eſſet purè violentus, vt putat Tartalea, corpus illud minimè de-
                    <lb/>
                  ſcenderet, eo quod uirtus mouens, in
                    <var>.a.</var>
                  poſita, nullo pacto poteſt talem effectum ef-
                    <lb/>
                  ficere, vnde ab ipſa natura prouenit deſcenſio illius corporis propter
                    <reg norm="grauitatem" type="context">grauitatẽ</reg>
                  ,
                    <reg norm="quam" type="context">quã</reg>
                    <lb/>
                  dictum corpus habet in tali medio, aeris ſcilicet, & non ex violentia aliqua. </s>
                  <s xml:id="echoid-s4362" xml:space="preserve">Sed ſi
                    <lb/>
                  dixiſſet ipſe, illum motum eſſe purum naturalem, hoc eſſet falſum, eo quod purus
                    <lb/>
                  naturalis motus alicuius corporis non impediti, extra locum ſuum, ſit per lineam re
                    <lb/>
                  ctam, & non per curuam, vt videre eſt inter
                    <var>.c.</var>
                  et
                    <var>.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4363" xml:space="preserve">In vltima propoſitione deinde eiuſdem lib. quæ .6. eſt decipitur ſimiliter, & hæc
                    <lb/>
                  deceptio oritur ab ignoratione quintæ, & à putando motum naturalem non eſſe cau
                    <lb/>
                  ſam ipſius deſcenſus per ſpatium
                    <var>.c.d</var>
                  . </s>
                  <s xml:id="echoid-s4364" xml:space="preserve">Sed quia tibi ſignificaui expeditiorem viam
                    <lb/>
                  repeririad
                    <reg norm="cognoſcendam" type="context">cognoſcendã</reg>
                  proportionem inter
                    <var>.a.h.</var>
                  et
                    <var>.a.e.</var>
                  in vltima propoſitione ſe-
                    <lb/>
                  cundi lib. ipſius Tartaleæ, ipſam nunc tibi ſcribo. </s>
                  <s xml:id="echoid-s4365" xml:space="preserve">
                    <reg norm="Nam" type="context">Nã</reg>
                  iam ſcis angulum
                    <var>.h.l.i.</var>
                  diui-
                    <lb/>
                  ſum eſſe per æqualia ab
                    <var>.P.l.</var>
                  & quod
                    <var>.a.h.</var>
                  et
                    <var>.h.p.</var>
                  ęquales inuicem ſunt ex .6. primi Eu-
                    <lb/>
                  cli. </s>
                  <s xml:id="echoid-s4366" xml:space="preserve">vnde
                    <var>.p.i.</var>
                  et
                    <var>.a.h.</var>
                  æquales erunt inuicem ſimiliter, ſed ex .3. ſexti ita eſt ipſius
                    <var>.a.l.</var>
                    <lb/>
                  ad
                    <var>.l.i.</var>
                  vt ipſius
                    <var>.a.p.</var>
                  ad
                    <var>.p.i.</var>
                  & coniunctim ita erit
                    <var>.a.l.i.</var>
                  ad
                    <var>.l.i.</var>
                  vt
                    <var>.a.i.</var>
                  ad
                    <var>.p.i.</var>
                  ſed
                    <var>.a.l.</var>
                  cogni
                    <lb/>
                  ta eſt ex eius quadrato, et
                    <var>.l.i.</var>
                  etiam, cum æqualis ſit ipſi
                    <var>.a.i.</var>
                  vnde ex regula de tribus
                    <lb/>
                  notam habebimus
                    <var>.p.i.</var>
                  reſpectu
                    <var>.a.i.</var>
                  & ita reſpectu
                    <var>.a.e.</var>
                  ſi hypotheſes ipſius Tartaleæ
                    <lb/>
                  veræ ſunt.</s>
                </p>
              </div>
              <div xml:id="echoid-div710" type="letter" level="4" n="2">
                <head xml:id="echoid-head538" style="it" xml:space="preserve">Alia demonstratio impoßibilitatis diuidendi per æqualia
                  <lb/>
                proportionem ſuperparticularem in
                  <lb/>
                diſcretis.</head>
                <head xml:id="echoid-head539" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s4367" xml:space="preserve">QVod à me poſtulas, hoc eſt ſcientiam impoſſibilitatis diuidendi per æqualia
                    <lb/>
                  proportionem ſuperparticularem in numeris ſatis à Campano in .8. octaui
                    <lb/>
                  potes habere, Iacobus Faber Stapulenſis etiam idem tractat
                    <unsure/>
                  in libello ſuę muſicæ
                    <lb/>
                  demonſtratæ. </s>
                  <s xml:id="echoid-s4368" xml:space="preserve">Sed ſi etiam alia via idem deſideras, quamuis longiori, nih
                    <unsure/>
                  ilomi-
                    <lb/>
                  nus vniuerſaliori, conſidera duos numeros
                    <var>.g.</var>
                  et
                    <var>.h.</var>
                  inuicem relatos ſecundum pro-
                    <lb/>
                  portionem ſuperparcicularem, quam volueris. </s>
                  <s xml:id="echoid-s4369" xml:space="preserve">Tunc dico impoſſibile eſle, vt per
                    <lb/>
                  æqualia diuidatur, quod ſi dixeris poſſibile eſſe, ſit per te
                    <var>.K.</var>
                  medius numerus </s>
                </p>
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