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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            <s xml:id="echoid-s14830" xml:space="preserve">
              <pb o="212" file="0218" n="218" rhead="ALHAZEN"/>
            linea o z ſecat ſuperficiem, contingentem pyramidem, trãſeuntem per a z:</s>
            <s xml:id="echoid-s14831" xml:space="preserve"> linea ergo a z eſt ſub dif-
              <lb/>
            ferentia communi inter ſuperficiem o z h & ſuperficiem contingentem.</s>
            <s xml:id="echoid-s14832" xml:space="preserve"> Et hæc differentia conti-
              <lb/>
            net cum linea h z angulum rectum, [per fabricationem.</s>
            <s xml:id="echoid-s14833" xml:space="preserve">] Angulus ergo e z h obtuſus:</s>
            <s xml:id="echoid-s14834" xml:space="preserve"> ergo angu-
              <lb/>
            lus f z h acutus [per 13 p 1.</s>
            <s xml:id="echoid-s14835" xml:space="preserve">] Ponatur ergo in z f punctum f:</s>
            <s xml:id="echoid-s14836" xml:space="preserve"> à quo extrahatur perpendicularis f e ſu-
              <lb/>
            per a e:</s>
            <s xml:id="echoid-s14837" xml:space="preserve"> & extrahatur rectè.</s>
            <s xml:id="echoid-s14838" xml:space="preserve"> Concurret ergo cum linea a o:</s>
            <s xml:id="echoid-s14839" xml:space="preserve"> [per 11 ax.</s>
            <s xml:id="echoid-s14840" xml:space="preserve">] nam angulus o a e eſt acutus
              <lb/>
            [per theſin, & ad e rectus eſt.</s>
            <s xml:id="echoid-s14841" xml:space="preserve">] Concurrat ergo in n.</s>
            <s xml:id="echoid-s14842" xml:space="preserve"> Et extrahatur ex e linea e d æquidiſtans z h li-
              <lb/>
            neæ [per 17 p 3.</s>
            <s xml:id="echoid-s14843" xml:space="preserve">] Erit ergo [per 8 p 11] e d perpendicularis ſuper ſuperficiem, contingentem pyra-
              <lb/>
            midem, tranſeuntem per a e:</s>
            <s xml:id="echoid-s14844" xml:space="preserve"> & extrahatur ex e linea æquidiſtans lineæ z m:</s>
            <s xml:id="echoid-s14845" xml:space="preserve"> & ſit e l.</s>
            <s xml:id="echoid-s14846" xml:space="preserve"> Et extrahatur
              <lb/>
            ſuperficies, in qua ſunt lineæ l e, e d.</s>
            <s xml:id="echoid-s14847" xml:space="preserve"> Secabit ergo ſuperficiem pyramidis, & faciet ſectionem [per
              <lb/>
            5 th.</s>
            <s xml:id="echoid-s14848" xml:space="preserve"> 1.</s>
            <s xml:id="echoid-s14849" xml:space="preserve"> con.</s>
            <s xml:id="echoid-s14850" xml:space="preserve"> Apoll.</s>
            <s xml:id="echoid-s14851" xml:space="preserve">] Nam hæc ſuperficies eſt obliqua ſuper axem a d.</s>
            <s xml:id="echoid-s14852" xml:space="preserve"> Sit ergo ſectio d e c:</s>
            <s xml:id="echoid-s14853" xml:space="preserve"> & m z eſt
              <lb/>
            perpẽdicularis ſuper ſuperficiem
              <lb/>
              <figure xlink:label="fig-0218-01" xlink:href="fig-0218-01a" number="188">
                <variables xml:id="echoid-variables177" xml:space="preserve">a o u p m h z t x b n y c q s l d g e K f r</variables>
              </figure>
            a z h:</s>
            <s xml:id="echoid-s14854" xml:space="preserve"> & hoc declaratũ eſt in præ-
              <lb/>
            dictis.</s>
            <s xml:id="echoid-s14855" xml:space="preserve"> [præcedente numero, per
              <lb/>
            lemma ad 37 theor.</s>
            <s xml:id="echoid-s14856" xml:space="preserve"> opticor.</s>
            <s xml:id="echoid-s14857" xml:space="preserve"> Eucli
              <lb/>
            dis.</s>
            <s xml:id="echoid-s14858" xml:space="preserve">] Ergo linea l e eſt perpendi-
              <lb/>
            cularis ſuper ſuperficiẽ a e d [per
              <lb/>
            8 p 11.</s>
            <s xml:id="echoid-s14859" xml:space="preserve">] Ergo angulus a e l eſt re-
              <lb/>
            ctus.</s>
            <s xml:id="echoid-s14860" xml:space="preserve"> Et ſimiliter angulus a e d re-
              <lb/>
            ctus eſt [per 29 p 1] & a e n ſimili-
              <lb/>
            ter rectus.</s>
            <s xml:id="echoid-s14861" xml:space="preserve"> Ergo [per 5 p 11] lineæ
              <lb/>
            l e, n e, d e ſunt in eadem ſuperfi-
              <lb/>
            cie.</s>
            <s xml:id="echoid-s14862" xml:space="preserve"> Ergo linea fen eſt in ſuperfi-
              <lb/>
            cie ſectionis.</s>
            <s xml:id="echoid-s14863" xml:space="preserve"> Et extrahatur ex f li-
              <lb/>
            nea æquidiſtans lineæ d e:</s>
            <s xml:id="echoid-s14864" xml:space="preserve"> [per 31
              <lb/>
            p 1] & ſit f r.</s>
            <s xml:id="echoid-s14865" xml:space="preserve"> Hęc ergo linea æqui-
              <lb/>
            diſtat lineæ h z [per 30 p 1.</s>
            <s xml:id="echoid-s14866" xml:space="preserve">] Et
              <lb/>
            extrahatur ex z in ſuperficie o z h,
              <lb/>
            linea continens cum z t angulum,
              <lb/>
            æqualem angulo o z t.</s>
            <s xml:id="echoid-s14867" xml:space="preserve"> [per 23 p 1.</s>
            <s xml:id="echoid-s14868" xml:space="preserve">]
              <lb/>
            Hæc ergo linea concurret cum f r [per lemma Procli ad 29 p 1] quia ſecat z h æquidiſtantem f r:</s>
            <s xml:id="echoid-s14869" xml:space="preserve"> &
              <lb/>
            eſt in ſuperficie eius:</s>
            <s xml:id="echoid-s14870" xml:space="preserve"> quia z f eſt in ſuperficie eius [per 35 d 1.</s>
            <s xml:id="echoid-s14871" xml:space="preserve">] Concurrat ergo in r.</s>
            <s xml:id="echoid-s14872" xml:space="preserve"> Ergo duo an-
              <lb/>
            guli, qui ſunt apud r, f, ſunt æquales:</s>
            <s xml:id="echoid-s14873" xml:space="preserve"> ſunt enim æquales duobus angulis, qui ſunt apud z [nam per
              <lb/>
            29 p 1 o z t, z f r:</s>
            <s xml:id="echoid-s14874" xml:space="preserve"> item t z r, z r f æquantur.</s>
            <s xml:id="echoid-s14875" xml:space="preserve">] Duæ ergo lineæ r z, f z ſunt æquales [per 6 p 1.</s>
            <s xml:id="echoid-s14876" xml:space="preserve">] Et de-
              <lb/>
            claratum eſt, quòd linea f e n eſt in ſuperficie ſectionis:</s>
            <s xml:id="echoid-s14877" xml:space="preserve"> & linea f r eſt æquidiſtans e d:</s>
            <s xml:id="echoid-s14878" xml:space="preserve"> eſt ergo
              <lb/>
            in ſuperficie ſectionis [per 35 d 1.</s>
            <s xml:id="echoid-s14879" xml:space="preserve">] Et continuemus r e:</s>
            <s xml:id="echoid-s14880" xml:space="preserve"> erit ergo [per 7 p 11] in ſuperficie ſectio-
              <lb/>
            nis:</s>
            <s xml:id="echoid-s14881" xml:space="preserve"> & extrahatur d e ad k.</s>
            <s xml:id="echoid-s14882" xml:space="preserve"> Et declaratum eſt, quòd e a eſt perpendicularis ſuper ſuperficiem ſe-
              <lb/>
            ctionis:</s>
            <s xml:id="echoid-s14883" xml:space="preserve"> uterque ergo angulorum a e r, a e f rectus eſt:</s>
            <s xml:id="echoid-s14884" xml:space="preserve"> [per 3 d 11] & duæ lineæ f z, r z ſunt ęqua-
              <lb/>
            les [per concluſionem.</s>
            <s xml:id="echoid-s14885" xml:space="preserve">] Ergo duæ lineæ r e, f e ſunt ęquales.</s>
            <s xml:id="echoid-s14886" xml:space="preserve"> [Quia enim anguli a e r, a e f ſunt
              <lb/>
            recti:</s>
            <s xml:id="echoid-s14887" xml:space="preserve"> quadrata z e, e f æquantur quadrato z f per 47 p 1:</s>
            <s xml:id="echoid-s14888" xml:space="preserve"> item q́ue quadrata z e, e r quadrato z r:</s>
            <s xml:id="echoid-s14889" xml:space="preserve">
              <lb/>
            at quadrata laterum z f, z r æqualium æquantur:</s>
            <s xml:id="echoid-s14890" xml:space="preserve"> quare ablato communi quadrato z e:</s>
            <s xml:id="echoid-s14891" xml:space="preserve"> quadrata
              <lb/>
            e f, e r, ideo q́ue latera e f, e r æquabuntur.</s>
            <s xml:id="echoid-s14892" xml:space="preserve">] Ergo [per 5 p 1] duo anguli e r f, e f r ſunt æqua-
              <lb/>
            les.</s>
            <s xml:id="echoid-s14893" xml:space="preserve"> Ergo forma n reflectetur ad r ex e:</s>
            <s xml:id="echoid-s14894" xml:space="preserve"> [per 12 n 4:</s>
            <s xml:id="echoid-s14895" xml:space="preserve"> quia anguli n e k, r e k æquantur, cum per
              <lb/>
            29 p 1 æquentur æqualibus ad f & r] & forma o reflectetur ad r ex z.</s>
            <s xml:id="echoid-s14896" xml:space="preserve"> Et omnis linea extracta ex
              <lb/>
            f ad aliquod punctum lineæ o n, ſecabit a e.</s>
            <s xml:id="echoid-s14897" xml:space="preserve"> Et patet, quòd linea illa erit æqualis lineæ extractæ
              <lb/>
            ex r ad idem punctum.</s>
            <s xml:id="echoid-s14898" xml:space="preserve"> Nam a e eſt perpendicularis ſuper ſuperficiem, in qua ſunt lineæ r e, f e:</s>
            <s xml:id="echoid-s14899" xml:space="preserve">
              <lb/>
            nam hæc ſuperficies eſt ſuperficies ſectionis:</s>
            <s xml:id="echoid-s14900" xml:space="preserve"> & duæ lineæ r e, f e ſunt æquales.</s>
            <s xml:id="echoid-s14901" xml:space="preserve"> Ergo omnes duæ
              <lb/>
            lineæ extractæ ex r, f ad unum aliquod punctum lineæ a e, ſunt æquales.</s>
            <s xml:id="echoid-s14902" xml:space="preserve"> Patet ergo, quòd forma
              <lb/>
            puncti, quod eſt in o n, reflectetur ad r exillo puncto, quod ſecatur in z e.</s>
            <s xml:id="echoid-s14903" xml:space="preserve"> Et ſimiliter de omni
              <lb/>
            puncto poſito in a n ultra n, ſi copulatum fuerit cum f per lineam rectam, illa linea ſecabit a e ul-
              <lb/>
            tra e.</s>
            <s xml:id="echoid-s14904" xml:space="preserve"> Patet ergo ex hoc, quòd forma lineæ a n, & quicquid continuatur cum ipſa, reflectetur ad
              <lb/>
            r à ſuperficie pyramidis a b g ex linea recta.</s>
            <s xml:id="echoid-s14905" xml:space="preserve"> Et ſimiliter omnis linea extracta ex a, obliqua ſuper
              <lb/>
            axem.</s>
            <s xml:id="echoid-s14906" xml:space="preserve"> Et continuemus n d:</s>
            <s xml:id="echoid-s14907" xml:space="preserve"> ſecabit ergo circumferentiam ſectionis:</s>
            <s xml:id="echoid-s14908" xml:space="preserve"> nam duo puncta d, n ſunt in
              <lb/>
            ſuperficie ſectionis, & n eſt extra circumferentiam ſectionis:</s>
            <s xml:id="echoid-s14909" xml:space="preserve"> & d eſt intra ſectionem.</s>
            <s xml:id="echoid-s14910" xml:space="preserve"> Secet ergo
              <lb/>
            circũferẽtiã ſectionis in c.</s>
            <s xml:id="echoid-s14911" xml:space="preserve"> Et quia triangulũ a o h eſt in eadẽ ſuperficie [per 2 p 11] erit [per 1 p 11] n d
              <lb/>
            in ſuperficie trianguli a o h:</s>
            <s xml:id="echoid-s14912" xml:space="preserve"> c ergo eſt in ſuperficie trianguli a o h:</s>
            <s xml:id="echoid-s14913" xml:space="preserve"> & duo puncta a, u ſunt in ſu-
              <lb/>
            perficie trianguli huius a o h:</s>
            <s xml:id="echoid-s14914" xml:space="preserve"> ſed puncta a, u, c ſunt in ſuperficie pyramidis.</s>
            <s xml:id="echoid-s14915" xml:space="preserve"> Ergo puncta a, u, c ſunt
              <lb/>
            in differentia communi ſuperficiei pyramidis, & ſuperficiei a u d:</s>
            <s xml:id="echoid-s14916" xml:space="preserve"> ſed hæc differentia eſt linea re-
              <lb/>
            cta [per 18 d 11.</s>
            <s xml:id="echoid-s14917" xml:space="preserve">] Ergo puncta a, u, c ſunt in linea recta.</s>
            <s xml:id="echoid-s14918" xml:space="preserve"> Extrahatur ergo a u rectè ad c:</s>
            <s xml:id="echoid-s14919" xml:space="preserve"> & extra-
              <lb/>
            hatur r z rectè:</s>
            <s xml:id="echoid-s14920" xml:space="preserve"> ſecabit ergo o h [quia ſecat angulum z h o baſi h o ſubtenſum, & utraque z r & h o
              <lb/>
            ſunt in uno plano.</s>
            <s xml:id="echoid-s14921" xml:space="preserve">] Secet ergo in puncto p.</s>
            <s xml:id="echoid-s14922" xml:space="preserve"> Eſt ergo p in ſuperficie trianguli a o h.</s>
            <s xml:id="echoid-s14923" xml:space="preserve"> Continuetur
              <lb/>
            ergo a p, & tranſeat rectè.</s>
            <s xml:id="echoid-s14924" xml:space="preserve"> Secabit ergo n d in g [quia ſecat angulum d a n.</s>
            <s xml:id="echoid-s14925" xml:space="preserve">] Et quia f non eſt in ſu-
              <lb/>
            perficie pyramidẽ contingente, trãſeunte per lineã a z:</s>
            <s xml:id="echoid-s14926" xml:space="preserve"> [ex concluſo] erit angulus fe d acutus.</s>
            <s xml:id="echoid-s14927" xml:space="preserve"> [Nã
              <lb/>
            quia per concluſionem punctũ f eſt in plano ſectionis ſeu ellipſis, obliquo ad a d e planũ axis, per 5
              <lb/>
            th.</s>
            <s xml:id="echoid-s14928" xml:space="preserve"> 1 con.</s>
            <s xml:id="echoid-s14929" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s14930" xml:space="preserve"> & angulus a e frectus eſt cõcluſus:</s>
            <s xml:id="echoid-s14931" xml:space="preserve"> erit angulus f e d acutus:</s>
            <s xml:id="echoid-s14932" xml:space="preserve"> & angulus d e n eſt obtu
              <lb/>
            </s>
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