Ibn-al-Haitam, al-Hasan Ibn-al-Hasan; Witelo; Risner, Friedrich, Opticae thesavrvs Alhazeni Arabis libri septem, nunc primùm editi. Eivsdem liber De Crepvscvlis & Nubium ascensionibus. Item Vitellonis Thuvringopoloni Libri X. Omnes instaurati, figuris illustrati & aucti, adiectis etiam in Alhazenum commentarijs, a Federico Risnero, 1572

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          <p>
            <s xml:id="echoid-s8989" xml:space="preserve">
              <pb o="148" file="0154" n="154" rhead="ALHAZEN"/>
            & t u a ſunt minores duobus rectis [per 17 p 1] cum a t & t u concurrant.</s>
            <s xml:id="echoid-s8990" xml:space="preserve"> Quare duo anguli t u a & d
              <lb/>
            g u ſunt minores duobus rectis:</s>
            <s xml:id="echoid-s8991" xml:space="preserve"> igitur [per 11 ax.</s>
            <s xml:id="echoid-s8992" xml:space="preserve">] a u concurret cum d g.</s>
            <s xml:id="echoid-s8993" xml:space="preserve"> Dico, quòd concurret in
              <lb/>
            puncto d:</s>
            <s xml:id="echoid-s8994" xml:space="preserve"> quoniam efficiet cum lineis u g, g d triangulum ſimile triãgulo a u t:</s>
            <s xml:id="echoid-s8995" xml:space="preserve"> habebunt enim angu
              <lb/>
            lum a u g communem:</s>
            <s xml:id="echoid-s8996" xml:space="preserve"> & angulus t a u eſt æqualis angulo d g u [per concluſionẽ.</s>
            <s xml:id="echoid-s8997" xml:space="preserve">] Igitur [per 4 p 6]
              <lb/>
            proportio a u ad a t, ſicut u g ad lineam, quã ſecat a u ex d g:</s>
            <s xml:id="echoid-s8998" xml:space="preserve"> & [per 3 p 6] proportio e a ad a u, ſicute
              <lb/>
            gad g u:</s>
            <s xml:id="echoid-s8999" xml:space="preserve"> cũ ſit angulus u a g ęqualis angulo g a e [per fabricationem.</s>
            <s xml:id="echoid-s9000" xml:space="preserve">] Cum ergo eadem ſit propor-
              <lb/>
            tio e a ad a t, ficut e g ad g d [ex concluſo] & proportio e a ad a t, ſit compoſita ex proportione e a ad
              <lb/>
            a u & a u ad a t [ratio enim extremorum cõponitur ex omnib.</s>
            <s xml:id="echoid-s9001" xml:space="preserve"> rationibus intermedijs, ut demonſtra
              <lb/>
            uit Theon ad 5 d 6] erit proportio e g ad g d cõpoſita ex ijſdem.</s>
            <s xml:id="echoid-s9002" xml:space="preserve"> Quare erit cõpacta ex proportione
              <lb/>
            e g ad g u & g u ad lineã, quã ſecat a u ex d g.</s>
            <s xml:id="echoid-s9003" xml:space="preserve"> Sed [ratio e g ad g d] eſt cõpacta ex proportionib.</s>
            <s xml:id="echoid-s9004" xml:space="preserve"> e g ad
              <lb/>
            g u & g u ad g d.</s>
            <s xml:id="echoid-s9005" xml:space="preserve"> Igitur linea, quã ſecat a u ex g d, eſt linea g d:</s>
            <s xml:id="echoid-s9006" xml:space="preserve"> igitur a u ſecat d g in puncto d.</s>
            <s xml:id="echoid-s9007" xml:space="preserve"> Produca
              <lb/>
            tur ergo [per 17 p 3] à puncto a cõtingens:</s>
            <s xml:id="echoid-s9008" xml:space="preserve"> quę ſit h a:</s>
            <s xml:id="echoid-s9009" xml:space="preserve"> erit ergo [per 18 p 3] g a h rectus:</s>
            <s xml:id="echoid-s9010" xml:space="preserve"> ſed g a l eſt me
              <lb/>
            dietas anguli d g u:</s>
            <s xml:id="echoid-s9011" xml:space="preserve"> igitur angulus l a h eſt medietas anguli d g e:</s>
            <s xml:id="echoid-s9012" xml:space="preserve"> cũ illi duo [d g u, d g e] ualeãt duos
              <lb/>
            rectos [per 13 p 1.</s>
            <s xml:id="echoid-s9013" xml:space="preserve">] Sed cũ angulus t a u ſit æqualis angulo d g u:</s>
            <s xml:id="echoid-s9014" xml:space="preserve"> erit angulus t a d ęqualis d g e [per 13
              <lb/>
            p 1.</s>
            <s xml:id="echoid-s9015" xml:space="preserve"> 3 ax.</s>
            <s xml:id="echoid-s9016" xml:space="preserve">] Igitur angulus l a h eſt medietas anguli t a d:</s>
            <s xml:id="echoid-s9017" xml:space="preserve"> & angulus e a l medietas anguli e a t [quia, ut
              <lb/>
            patuit, e a l æquatur ipſi l a t:</s>
            <s xml:id="echoid-s9018" xml:space="preserve">] igitur angulus e a h medietas anguli e a d.</s>
            <s xml:id="echoid-s9019" xml:space="preserve"> Quare a h diuidit angulum
              <lb/>
            e a d per ęqualia.</s>
            <s xml:id="echoid-s9020" xml:space="preserve"> Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s9021" xml:space="preserve"> Si uerò a u (cum ſit angulus ſuper punctum a ęqualis angu
              <lb/>
            lo g a e) non cadit ſuper lineam e s extra circulum, uel intra:</s>
            <s xml:id="echoid-s9022" xml:space="preserve"> ſit ergo æquidiſtans.</s>
            <s xml:id="echoid-s9023" xml:space="preserve"> Igitur [ք 29 p 1] an
              <lb/>
            gulus u a g ęqualis eſt angulo a g e:</s>
            <s xml:id="echoid-s9024" xml:space="preserve"> ſed idem eſt æqualis angulo g a e [ex theſi.</s>
            <s xml:id="echoid-s9025" xml:space="preserve">] Quare [per 1 ax.</s>
            <s xml:id="echoid-s9026" xml:space="preserve">] an
              <lb/>
            gulus g a e eſt æqua-
              <lb/>
              <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a" number="76">
                <variables xml:id="echoid-variables66" xml:space="preserve">d a u m l t z c g s h q</variables>
              </figure>
            lis angulo a g e:</s>
            <s xml:id="echoid-s9027" xml:space="preserve"> igi-
              <lb/>
            tur [per 6 p 1] e g eſt
              <lb/>
            æqualis a e.</s>
            <s xml:id="echoid-s9028" xml:space="preserve"> Simili
              <lb/>
            ter angulus t a d erit
              <lb/>
            ęqualis angulo a t g
              <lb/>
            [per 29 p 1.</s>
            <s xml:id="echoid-s9029" xml:space="preserve">] Sed iam
              <lb/>
            dictum eſt [in primo
              <lb/>
            caſu huius numeri]
              <lb/>
            quòd angulus t a d
              <lb/>
            eſt ęqualis angulo d
              <lb/>
            g t.</s>
            <s xml:id="echoid-s9030" xml:space="preserve"> Igitur angulus a t g eſt ęqualis angulo d g t:</s>
            <s xml:id="echoid-s9031" xml:space="preserve"> & ſimiliter [per 29 p 1] duo anguli a d g, d g t ſunt ę-
              <lb/>
            quales:</s>
            <s xml:id="echoid-s9032" xml:space="preserve"> igitur duo anguli a d g, a t g ſunt ęquales.</s>
            <s xml:id="echoid-s9033" xml:space="preserve"> Sequetur ergo ex his, quòd linea, quam ſecat a u ex
              <lb/>
            d g, ſit ęqualis lineæ a t [nam cũ anguli a t g, d g t:</s>
            <s xml:id="echoid-s9034" xml:space="preserve"> itẽ a d g, t a d ęquentur:</s>
            <s xml:id="echoid-s9035" xml:space="preserve"> ęquabitur per 6 p 1 t m ipſi
              <lb/>
            m g:</s>
            <s xml:id="echoid-s9036" xml:space="preserve"> item m d ipſi m a.</s>
            <s xml:id="echoid-s9037" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s9038" xml:space="preserve"> ſi ęqualibus ęqualia addantur:</s>
            <s xml:id="echoid-s9039" xml:space="preserve"> ęquabitur d g ipſi a t.</s>
            <s xml:id="echoid-s9040" xml:space="preserve">] Et iam dictum eſt,
              <lb/>
            quòd e g ęqualis ſit a e.</s>
            <s xml:id="echoid-s9041" xml:space="preserve"> Igitur [per 7 p 5] proportio e g ad lineam, quã ſecat a u e x d g, eſt ſicut a e ad
              <lb/>
            a t.</s>
            <s xml:id="echoid-s9042" xml:space="preserve"> Sed iam dictum eſt ut a e ad a t, ſic e g ad g d:</s>
            <s xml:id="echoid-s9043" xml:space="preserve"> igitur linea, quã ſecat a u ex d g, eſt d g.</s>
            <s xml:id="echoid-s9044" xml:space="preserve"> Et cum t a d
              <lb/>
            ſit æqualis d g t:</s>
            <s xml:id="echoid-s9045" xml:space="preserve"> erit l a h medietas anguli t a d, ſicut dictum eſt ſuprà, & e a l medietas e a t.</s>
            <s xml:id="echoid-s9046" xml:space="preserve"> Erit ergo
              <lb/>
            e a h medietas anguli e a d.</s>
            <s xml:id="echoid-s9047" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s9048" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div341" type="section" level="0" n="0">
          <head xml:id="echoid-head340" xml:space="preserve" style="it">37. À
            <unsure/>
          dato extra circulum puncto, ducere ad datam diametrũ, lineã rectã: cui{us} pars inter
            <lb/>
          peripheriam & datam diametrum æquetur parti diametri centro circuli conterminæ. 136 p 1.</head>
          <p>
            <s xml:id="echoid-s9049" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s9050" xml:space="preserve"> dato circulo, cuius centrum g:</s>
            <s xml:id="echoid-s9051" xml:space="preserve"> & data in eo diametro b g:</s>
            <s xml:id="echoid-s9052" xml:space="preserve"> & dato e puncto extra cir-
              <lb/>
            culum:</s>
            <s xml:id="echoid-s9053" xml:space="preserve"> eſt ducere à puncto e ad diametrum b g, lineã ſecãtem circulum, ita ut pars eius à cir-
              <lb/>
            culo uſq;</s>
            <s xml:id="echoid-s9054" xml:space="preserve"> ad diametrũ ſit ęqualis parti diametri, interiacenti inter ipſam & centrum.</s>
            <s xml:id="echoid-s9055" xml:space="preserve"> Verbi
              <lb/>
            gratia:</s>
            <s xml:id="echoid-s9056" xml:space="preserve"> ducatur à puncto e perpendicularis ſuper diametrum:</s>
            <s xml:id="echoid-s9057" xml:space="preserve"> & ſit e c:</s>
            <s xml:id="echoid-s9058" xml:space="preserve"> & ducatur linea e g:</s>
            <s xml:id="echoid-s9059" xml:space="preserve"> & ſuma-
              <lb/>
            tur linea q t æqualis e c:</s>
            <s xml:id="echoid-s9060" xml:space="preserve"> & [per 33 p 3] fiat ſuper q t portio circuli, ut quilibet angulus cadens in hanc
              <lb/>
            portionem, ſit ęqualis angul
              <gap/>
            e g b:</s>
            <s xml:id="echoid-s9061" xml:space="preserve"> & compleatur circulus [per 25 p 3] & à medio puncto q t duca-
              <lb/>
            tur ex utraq;</s>
            <s xml:id="echoid-s9062" xml:space="preserve"> parte perpẽdicularis uſq;</s>
            <s xml:id="echoid-s9063" xml:space="preserve"> ad circulũ:</s>
            <s xml:id="echoid-s9064" xml:space="preserve"> erit quidẽ [per coſectarium 1 p 3] dιameter huius
              <lb/>
            circuli:</s>
            <s xml:id="echoid-s9065" xml:space="preserve"> & à puncto q ducatur linea ad hanc diametrũ, ſecans eam in puncto f, & producatur uſq;</s>
            <s xml:id="echoid-s9066" xml:space="preserve"> ad
              <lb/>
            punctum p circuli, ita ut f p ſit æqualis medietati g b [per 34 n] & ducatur linea p t, & linea t f, Et du
              <lb/>
            catur à puncto p linea
              <lb/>
              <figure xlink:label="fig-0154-02" xlink:href="fig-0154-02a" number="77">
                <variables xml:id="echoid-variables67" xml:space="preserve">k b d z e i c g x</variables>
              </figure>
              <figure xlink:label="fig-0154-03" xlink:href="fig-0154-03a" number="78">
                <variables xml:id="echoid-variables68" xml:space="preserve">p
                  <gap/>
                n f o m u q ſ
                  <gap/>
                </variables>
              </figure>
            ęquidiſtans diametro:</s>
            <s xml:id="echoid-s9067" xml:space="preserve">
              <lb/>
            quæ ſit p u:</s>
            <s xml:id="echoid-s9068" xml:space="preserve"> cõcurratq́;</s>
            <s xml:id="echoid-s9069" xml:space="preserve">
              <lb/>
            cũ t f in puncto u:</s>
            <s xml:id="echoid-s9070" xml:space="preserve"> [con
              <lb/>
            curret autem per lem-
              <lb/>
            ma Procli ad 29 p 1] &
              <lb/>
            à puncto u ducatur æ-
              <lb/>
            quidiſtãs t q:</s>
            <s xml:id="echoid-s9071" xml:space="preserve"> quæ ſit u
              <lb/>
            o:</s>
            <s xml:id="echoid-s9072" xml:space="preserve"> & à pũcto t ducatur
              <lb/>
            perpendicularis ſuper
              <lb/>
            p q:</s>
            <s xml:id="echoid-s9073" xml:space="preserve"> quæ ſit t n:</s>
            <s xml:id="echoid-s9074" xml:space="preserve"> & à pun
              <lb/>
            cto t ducatur æquidiſtans p q:</s>
            <s xml:id="echoid-s9075" xml:space="preserve"> quæ ſit t s:</s>
            <s xml:id="echoid-s9076" xml:space="preserve"> & à puncto u perpendicularis ſuper p q:</s>
            <s xml:id="echoid-s9077" xml:space="preserve"> quæ ſit u h.</s>
            <s xml:id="echoid-s9078" xml:space="preserve"> Dein
              <lb/>
            de [per 23 p 1] ex angulo b g e ſecetur angulus æqualis angulo q p u:</s>
            <s xml:id="echoid-s9079" xml:space="preserve"> [id aũt fieri poteſt, cum totus
              <lb/>
            angulus q p t ęquetur ք theſin angulo b g e:</s>
            <s xml:id="echoid-s9080" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s9081" xml:space="preserve"> pars illius ab hoc toto detrahi poteſt] ꝗ ſit b g d:</s>
            <s xml:id="echoid-s9082" xml:space="preserve">
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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