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-s28542" xml:space="preserve">
              <pb o="142" file="0444" n="444" rhead="VITELLONIS OPTICAE"/>
            ſuperficiei circuli maior uel minor ſemidiametro d a:</s>
            <s xml:id="echoid-s28543" xml:space="preserve"> ſit tamẽ angulus d a z æqualis angulo g a z, &
              <lb/>
            angulus e a z æqualis angulo b a z.</s>
            <s xml:id="echoid-s28544" xml:space="preserve"> Dico, quòd adhuc diametri d b & e g uidebuntur æquales:</s>
            <s xml:id="echoid-s28545" xml:space="preserve"> quo-
              <lb/>
              <figure xlink:label="fig-0444-01" xlink:href="fig-0444-01a" number="484">
                <variables xml:id="echoid-variables464" xml:space="preserve">e z b a d g z</variables>
              </figure>
            niam enim linea d a eſt ęqualis a g, & linea z a communis duobus trigo-
              <lb/>
            nis z a g, & z a d:</s>
            <s xml:id="echoid-s28546" xml:space="preserve"> eſt quoq;</s>
            <s xml:id="echoid-s28547" xml:space="preserve"> ex hypotheſi angulus d a z æqualis angulo g
              <lb/>
            a z:</s>
            <s xml:id="echoid-s28548" xml:space="preserve"> erit per 4 p 1 linea z d æqualis lineæ z g, & angulus d z a æqualis an-
              <lb/>
            gulo g z a:</s>
            <s xml:id="echoid-s28549" xml:space="preserve"> ergo per 19 uel 20 huius baſis d a uidebitur æqualis g a baſi.</s>
            <s xml:id="echoid-s28550" xml:space="preserve">
              <lb/>
            Similiter quoq;</s>
            <s xml:id="echoid-s28551" xml:space="preserve"> per eadem demonſtrabitur angulus e z a æqualis angu-
              <lb/>
            lo b z a:</s>
            <s xml:id="echoid-s28552" xml:space="preserve"> & per pręmiſſa uidebitur linea e a ęqualis lineæ b a, & angulus a
              <lb/>
            z g æqualis eſt angulo a z d, & angulus e z a æqualis angulo a z g:</s>
            <s xml:id="echoid-s28553" xml:space="preserve"> ideo
              <lb/>
            accidit ut totalis angulus d z b totali angulo e z g ſit æqualis.</s>
            <s xml:id="echoid-s28554" xml:space="preserve"> Videbitur
              <lb/>
            ergo, ut ſuprà patuit, diameter d b æqualis diametro e g.</s>
            <s xml:id="echoid-s28555" xml:space="preserve"> Quod eſt pro-
              <lb/>
            poſitum.</s>
            <s xml:id="echoid-s28556" xml:space="preserve"> Poſsibile eſt autem hoc in quibuſdam diametris accidere, non
              <lb/>
            autem in omnibus diametris circuli taliter uiſui oppoſiti:</s>
            <s xml:id="echoid-s28557" xml:space="preserve"> nõ ergo opor-
              <lb/>
            tet quòd omnes diametri illius circuli uideantur æquales:</s>
            <s xml:id="echoid-s28558" xml:space="preserve"> non enim
              <lb/>
            illæ diametri uidebuntur æquales, cum quibus linea z a facit angulos
              <lb/>
            in æquales.</s>
            <s xml:id="echoid-s28559" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1134" type="section" level="0" n="0">
          <head xml:id="echoid-head903" xml:space="preserve" style="it">55. Sirect a linea à centro circuli centro oculi incidens, non eri-
            <lb/>
          gatur ſuper ſuperficiem circuli, ne æquales angulos contineat cum
            <lb/>
          diametris, ſit́ maior ſemidiametro: diametri illius circuliinæqua-
            <lb/>
          les apparebunt: totus́ circulus uidebitur ſectio columnaris: cuius
            <lb/>
          maxima eſt diameter illa, cui perpendiculariter incidit linea radia-
            <lb/>
          lis. Euclides 37. 39 th. opt.</head>
          <p>
            <s xml:id="echoid-s28560" xml:space="preserve">Eſto circulus a g b d:</s>
            <s xml:id="echoid-s28561" xml:space="preserve"> cuius centrum z:</s>
            <s xml:id="echoid-s28562" xml:space="preserve"> & ducantur diametri a b & g d, ſe ad inuicem orthogona
              <lb/>
            liter ſecantes:</s>
            <s xml:id="echoid-s28563" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s28564" xml:space="preserve"> centrum oculi e:</s>
            <s xml:id="echoid-s28565" xml:space="preserve"> à quo ducatur linea e z ad centrum circuli, diametro quidem d
              <lb/>
            g ſecundum angulum rectum perpendiculariter incidens, diametro uerò a b obliquè, ut acciderit:</s>
            <s xml:id="echoid-s28566" xml:space="preserve">
              <lb/>
            non erit ergo linea e z erecta ſuper ſuperficiem circuli:</s>
            <s xml:id="echoid-s28567" xml:space="preserve"> ſitq́ linea e z maior ſemidiametro circuli.</s>
            <s xml:id="echoid-s28568" xml:space="preserve">
              <lb/>
            Dico, quòd diametri a b & g d uidebuntur in æquales:</s>
            <s xml:id="echoid-s28569" xml:space="preserve"> & g d maxima quidem, a b uerò minima:</s>
            <s xml:id="echoid-s28570" xml:space="preserve"> &
              <lb/>
            quòd totus circulus uidebitur altera parte longior, ueluti ſectio columnaris:</s>
            <s xml:id="echoid-s28571" xml:space="preserve"> & quòd omnis dia-
              <lb/>
            meter circuli, quæ ceciderit propior minimæ, uidebitur minor remotiore ab illa:</s>
            <s xml:id="echoid-s28572" xml:space="preserve"> & duæ tãtùm dia-
              <lb/>
            metri apparebunt æquales, ut illæ, quæ æqualiter diſtant ab utraq;</s>
            <s xml:id="echoid-s28573" xml:space="preserve"> parte à minima diametro, quæ
              <lb/>
            eſt a b.</s>
            <s xml:id="echoid-s28574" xml:space="preserve"> Quoniam enim diameter g d eſt perpendicularis ſuper diametrum a b, & ſuper lineam z e,
              <lb/>
            palàm per 4 p 11 quoniam linea g z eſt perpendicularis ſuper ſuperficiem, in qua ſunt lineæ e z & a
              <lb/>
            z, uel a b:</s>
            <s xml:id="echoid-s28575" xml:space="preserve"> ergo per 18 p 11 erit circulus propoſitus orthogonalis ſuper ſuperficiem e a z:</s>
            <s xml:id="echoid-s28576" xml:space="preserve"> ergo & e a z
              <lb/>
            ſuperficies erecta erit ſuper circulum.</s>
            <s xml:id="echoid-s28577" xml:space="preserve"> Ducatur ergo à puncto e ſuper ſuperficiem circuli a b g d
              <lb/>
              <figure xlink:label="fig-0444-02" xlink:href="fig-0444-02a" number="485">
                <variables xml:id="echoid-variables465" xml:space="preserve">k e a i p g z d s b t</variables>
              </figure>
            perpendicularis per 11 p 11:</s>
            <s xml:id="echoid-s28578" xml:space="preserve"> hæc itaque per præmiſſa ne-
              <lb/>
            ceſſariò cadet in communem ſectionem illarum ſuper-
              <lb/>
            ficierum, quæ eſt a b:</s>
            <s xml:id="echoid-s28579" xml:space="preserve"> cadat ergo, & ſit e k:</s>
            <s xml:id="echoid-s28580" xml:space="preserve"> & ducantur li-
              <lb/>
            neæ e a, e b, e d, e g:</s>
            <s xml:id="echoid-s28581" xml:space="preserve"> producaturq́ diameter circuli alia,
              <lb/>
            quæ ſit s z p, conſtituens cum diametro g z d angulum
              <lb/>
            p z d æqualem angulo g z s per 15 p 1:</s>
            <s xml:id="echoid-s28582" xml:space="preserve"> ducatur quoque
              <lb/>
            alia diameter, quæ ſit i z t:</s>
            <s xml:id="echoid-s28583" xml:space="preserve"> ita ut anguli g z s & i z g ſint
              <lb/>
            æquales.</s>
            <s xml:id="echoid-s28584" xml:space="preserve"> Quia itaque à puncto e in aere dato ſuper ſub-
              <lb/>
            ſtratam planam ſuperficiem circuli, qui eſt a b g d, du-
              <lb/>
            cuntur duæ lineæ, una perpendiculariter, quæ eſt e k, &
              <lb/>
            alia obliquè, quæ eſt e z, & inter puncta incidentiæ, quæ
              <lb/>
            ſunt k & z, copulatur linea z k in ipſa ſuperſicie:</s>
            <s xml:id="echoid-s28585" xml:space="preserve"> patet
              <lb/>
            per 39 th.</s>
            <s xml:id="echoid-s28586" xml:space="preserve"> 1 huius, quoniam angulus e z k minimus eſt o-
              <lb/>
            mnium angulorum ſub linea e z obliquè incidente, &
              <lb/>
            ſemidiametro z i uel z p, uel quacunq;</s>
            <s xml:id="echoid-s28587" xml:space="preserve"> alia diametro con
              <lb/>
            tentorum:</s>
            <s xml:id="echoid-s28588" xml:space="preserve"> & omnis angulus iſtorum angulorum pro-
              <lb/>
            pinquior angulo e z k eſt minor remotiore:</s>
            <s xml:id="echoid-s28589" xml:space="preserve"> duo quo que
              <lb/>
            anguli ex utraque parte æqualiter angulo e z k appro-
              <lb/>
            ximantes, ut ſunt anguli i z k, & p z k inter ſe ſunt æqua-
              <lb/>
            les.</s>
            <s xml:id="echoid-s28590" xml:space="preserve"> Copulentur quoq;</s>
            <s xml:id="echoid-s28591" xml:space="preserve"> lineæ e i, e s, e p, e t.</s>
            <s xml:id="echoid-s28592" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s28593" xml:space="preserve"> ab
              <lb/>
            angulis duorũ trigonorũ d e g & t e i, ad medietates ſua-
              <lb/>
            rũ baſiũ æqualiũ in trigono d e g linea e z perpẽdicula-
              <lb/>
            riter incidit, & in trigono tei obliquè, eſtq́;</s>
            <s xml:id="echoid-s28594" xml:space="preserve"> linea e z ma
              <lb/>
            ior medietate utriuſq;</s>
            <s xml:id="echoid-s28595" xml:space="preserve"> illarũ baſium g d, & i t, ut patet ex
              <lb/>
            hypotheſi:</s>
            <s xml:id="echoid-s28596" xml:space="preserve"> ergo ք 49 th.</s>
            <s xml:id="echoid-s28597" xml:space="preserve"> 1 huius erit angulus d e g maior
              <lb/>
            angulo t e i:</s>
            <s xml:id="echoid-s28598" xml:space="preserve"> ergo ք 20 huius diameter d g uidebitur ma-
              <lb/>
            ior diametro i t.</s>
            <s xml:id="echoid-s28599" xml:space="preserve"> Et quoniã, ut oſtẽſum eſt ք 39 th.</s>
            <s xml:id="echoid-s28600" xml:space="preserve"> 1 huius,
              <lb/>
            angulus e z i eſt maior angulo e z a, ambabus uerò baſib.</s>
            <s xml:id="echoid-s28601" xml:space="preserve"> trigonorũ t e i & a e b, quæ ſunt i t & a b, ad
              <lb/>
            </s>
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