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-div639" type="section" level="3" n="27">
              <div xml:id="echoid-div639" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s4008" xml:space="preserve">
                    <pb o="331" rhead="EPISTOL AE." n="343" file="0343" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0343"/>
                  dabit .64. (nam duo triangula
                    <var>.V.S.*.</var>
                  et
                    <var>.V.E.ω.</var>
                  ſunt inuicem ſimilia eadem ratio-
                    <lb/>
                  ne) vnde
                    <var>.E.ω.</var>
                  veniet nobis extalibus partibus .462.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4009" xml:space="preserve">Coniungatur nunc quadratum ipſius
                    <var>.E.V.</var>
                  quod eſt .4096. cum quadrato ipſius
                    <var>.
                      <lb/>
                    E.ω.</var>
                  quod eſt .213444. & habebimus quadratum ipfius
                    <var>.V.ω.</var>
                  talium
                    <reg norm="partium" type="context">partiũ</reg>
                  .217540.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4010" xml:space="preserve">Dicemus poſtea ſi .217540. dat nobis .4096. quid dabit quadratum ipſius
                    <var>.V.ω.</var>
                  vt
                    <lb/>
                  ſinus totus quod eſt .10000000000. vnde veniet pro quadrato ipſius
                    <var>.V.E.</var>
                  talium
                    <lb/>
                  partium, ſuperficialium ſcilicet .18827211. cuius radix erit .13721. & erit ſinus an-
                    <lb/>
                  guli
                    <var>.V.ω.E.</var>
                  qui erit grad .7. min .53. vnde angulus
                    <var>.ω.V.E.</var>
                  erit grad .82. min .7. eius
                    <lb/>
                  vero ſinus erit partium .99054.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4011" xml:space="preserve">Nunc autem quia angulus
                    <var>.E.V.ω.</var>
                  eſt acutus, imaginemur
                    <var>.E.Ŕ.</var>
                  ductam eſſe ad re­
                    <lb/>
                  ctos ipſi
                    <var>.V.ω.</var>
                    <reg norm="ſitque" type="simple">ſitq́;</reg>
                  etiam ducta ipſa
                    <var>.E.G</var>
                  . </s>
                  <s xml:id="echoid-s4012" xml:space="preserve">Vnde habebimus angulum
                    <var>.Ŕ.E.V.</var>
                  gra-
                    <lb/>
                  duum .7. min .53. eius vero ſinus
                    <var>.Ŕ.V.</var>
                  partium .13721. (propter ſimilitudinem trian­
                    <lb/>
                  gulorum
                    <var>.E.Ŕ.V.</var>
                  et
                    <var>.ω.E.V.</var>
                  ) talium ſcilicet, qualium
                    <var>.E.V.</var>
                  fuerit .100000. </s>
                  <s xml:id="echoid-s4013" xml:space="preserve">Sed qua-
                    <lb/>
                  lium
                    <var>.E.V.</var>
                  eſt .64. talium erit .8. cum tribus quartis, cuius
                    <var>.Ŕ.V.</var>
                  quadratum erit par­
                    <lb/>
                  tium .76. cum dimidio ſimilium ſed ſuperſicialium, quo quidem quadrato dempto
                    <lb/>
                  ex quadrato ipſius .64. quod eſt .4096. remanebit quadratum ipſius
                    <var>.E.Ŕ.</var>
                  partium
                    <num value="2871">.
                      <lb/>
                    2871.</num>
                  quo etiam quadrato
                    <var>.E.Ŕ.</var>
                  dempto ex quadrato
                    <var>.E.G.</var>
                  partium .14400. remane­
                    <lb/>
                  bit quadratum ipſius
                    <var>.Ŕ.G.</var>
                  partium .11529. cuius radix
                    <var>.Ŕ.G.</var>
                  erit partium .107.
                    <reg norm="talium" type="context">taliũ</reg>
                    <lb/>
                  qualium
                    <var>.E.G.</var>
                  eſt .120. ſed qualium
                    <var>.E.G.</var>
                  erit .100000. talium
                    <var>.Ŕ.G.</var>
                  erit partium
                    <num value="89166">.
                      <lb/>
                    89166.</num>
                  quæ vt ſinus anguli
                    <var>.Ŕ.E.G.</var>
                  habebit pro ipſo angulo, gra .63. min .5. qui colle­
                    <lb/>
                  cti cum gra .7. min .53. anguli
                    <var>.V.E.Ŕ.</var>
                  dabunt totum angulum
                    <var>.A.E.G.</var>
                  grad .70.
                    <lb/>
                  min .58. cuius complementum ex grad .90. erit
                    <var>.G.D.</var>
                  graduum .19. min .2. & non .24.
                    <lb/>
                  vt omnes ferè putant.</s>
                </p>
              </div>
            </div>
            <div xml:id="echoid-div642" type="section" level="3" n="28">
              <div xml:id="echoid-div642" type="letter" level="4" n="1">
                <head xml:id="echoid-head495" xml:space="preserve">DE REFLEXIONIBVS RADIORVM.</head>
                <head xml:id="echoid-head496" style="it" xml:space="preserve">Excellentißimo Philoſopbo Franciſco Vimercato.</head>
                <p>
                  <s xml:id="echoid-s4014" xml:space="preserve">
                    <emph style="sc">QVoniam</emph>
                  non videbatur quieſcere animus tuus, cum paucis ab hinc die-
                    <lb/>
                  bus tibi ſiſcitanti reſpondiſſem, nec tamen rationem omnium, quæ dixe-
                    <lb/>
                  ram exactè explicare per tem poris anguſtiam potuiſſem, cogitaui ad te
                    <lb/>
                  per hanc occaſionem ſcribens, & iam dicta repetere, & omnium tibi ra-
                    <lb/>
                  tionem ſubiungere, & vt mihi plenius ſatisfaciam, & tibi commodè perlegenti faci
                    <lb/>
                  lius ſit veritatem intueri. </s>
                  <s xml:id="echoid-s4015" xml:space="preserve">Scripſiſti enim in tuis diſputationibus, vir doctiſſime, quod
                    <lb/>
                  omnis res viſa per
                    <reg norm="ſpeculum" type="context">ſpeculũ</reg>
                    <reg norm="quodcunque" type="context">quodcũque</reg>
                  , ſub breuiſſimis lineis
                    <reg norm="compræhendatur" type="context">cõpræhendatur</reg>
                  à vifu.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4016" xml:space="preserve">Propoſitio hæc non eſt vniuerſaliter vera (quamuis etiam ab alijs omnibus pro ta
                    <lb/>
                  li poſita ſit) cum in ſpeculis concauis non ſemper verificetur, vt nunc tibi demon-
                    <lb/>
                  ſtrabo.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4017" xml:space="preserve">Eſto quod linea recta
                    <var>.b.d.</var>
                  tangat circulum
                    <lb/>
                    <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a" number="367">
                      <image file="0343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0343-01"/>
                    </figure>
                    <var>b.o.q.n.</var>
                  qui ſit communis ſectionis ſup crficiei re
                    <lb/>
                  flexionis, & ſphæricę alicuius ſpeculi ſphærici
                    <lb/>
                  concaui, & punctum contingentiæ ſit
                    <var>.b.</var>
                  à quo
                    <lb/>
                  exeant duæ lineæ
                    <var>.b.q.</var>
                  et
                    <var>.b.n.</var>
                  efficientes duos an
                    <lb/>
                  gulos inuicem æquales circa perpendicularem
                    <var>.
                      <lb/>
                    b.c.</var>
                  res autem viſa primò ſit in ipſa circunferen-
                    <lb/>
                  tia huiuſmodi circuli in puncto
                    <var>.n.</var>
                  oculus vero in
                    <lb/>
                  puncto
                    <var>.q.</var>
                  ipſius circunferentię. </s>
                  <s xml:id="echoid-s4018" xml:space="preserve">Dico nunc duas </s>
                </p>
              </div>
            </div>
          </div>
        </div>
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