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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        <div xml:id="echoid-div412" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s11570" xml:space="preserve">
              <pb o="174" file="0180" n="180" rhead="ALHAZEN"/>
            ctis a, b ad idẽ pũctũ illius circuli extrà:</s>
            <s xml:id="echoid-s11571" xml:space="preserve"> fiet [per 21 p 1] angulus minor angulo a t b:</s>
            <s xml:id="echoid-s11572" xml:space="preserve"> & probabitur eſſe
              <lb/>
            æqualis.</s>
            <s xml:id="echoid-s11573" xml:space="preserve"> Quoniã [per 22 p 3] cũ angulo a g b ualebit duos rectos, & anguli a g b & a g d ualent duos
              <lb/>
            rectos:</s>
            <s xml:id="echoid-s11574" xml:space="preserve"> [per 13 p 1] & angulus a t b eſt æqualis angulo a g d ex hypotheſi:</s>
            <s xml:id="echoid-s11575" xml:space="preserve"> ergo angulus a t b cum angu
              <lb/>
            lo a g b ualet duos rectos.</s>
            <s xml:id="echoid-s11576" xml:space="preserve"> Et ita impoſsibile [cõtra 21 p 1.</s>
            <s xml:id="echoid-s11577" xml:space="preserve">] Similiter ſi circulus citra t ceciderit, eadẽ
              <lb/>
            erit improbatio.</s>
            <s xml:id="echoid-s11578" xml:space="preserve"> Reſtat ergo, ut tranſeat per punctum t.</s>
            <s xml:id="echoid-s11579" xml:space="preserve"> Cum igitur [per 12 n 4] angulus a t g ſit æqua
              <lb/>
            lis angulo b t g:</s>
            <s xml:id="echoid-s11580" xml:space="preserve"> erit [per 26 p 3] arcus a g æqualis arcui b g:</s>
            <s xml:id="echoid-s11581" xml:space="preserve"> & ita [per 29 p 3] a g erit æqualis b g:</s>
            <s xml:id="echoid-s11582" xml:space="preserve"> & po-
              <lb/>
            ſitum eſt eſſe eas inæquales.</s>
            <s xml:id="echoid-s11583" xml:space="preserve"> Et ita eſt propoſitum.</s>
            <s xml:id="echoid-s11584" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div414" type="section" level="0" n="0">
          <head xml:id="echoid-head383" xml:space="preserve" style="it">80. Si uiſus & uiſibile in diuerſis diametris circuli (qui eſt cõmunis ſectio ſuperficierũ, refle-
            <lb/>
          xionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia inter ſe reflectãtur à duobus pun
            <lb/>
          ctis peripheriæ, cõprehenſæ inter ſemidiametros, in quibus ipſa ſunt: nõ erit uter angulus cõpo
            <lb/>
          ſit us ex angulo incidẽtiæ & reflexionis, minor angulo exteriore à dictis diametris facto. 34 p 8.</head>
          <p>
            <s xml:id="echoid-s11585" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11586" xml:space="preserve"> ſumptis in duabus diametris e g h, z g d, duobus pũctis a, b, ut b g ſit maior a g.</s>
            <s xml:id="echoid-s11587" xml:space="preserve"> Dico,
              <lb/>
            quòd ſi punctũ a reflectatur ad b à duobus punctis arcus e z:</s>
            <s xml:id="echoid-s11588" xml:space="preserve"> nõ erit uterq;</s>
            <s xml:id="echoid-s11589" xml:space="preserve"> angulus conſtans
              <lb/>
            ex angulo incidentię & reflexiõis, minor angulo a g d.</s>
            <s xml:id="echoid-s11590" xml:space="preserve"> Sumãtur enim duo puncta t, q ιn arcu
              <lb/>
            e z, à quib.</s>
            <s xml:id="echoid-s11591" xml:space="preserve"> a reflectatur ad b:</s>
            <s xml:id="echoid-s11592" xml:space="preserve"> & ducãtur lineę b t, g t, a t, b q, g q, a q:</s>
            <s xml:id="echoid-s11593" xml:space="preserve"> & ſi angulus a t b minor eſt angu
              <lb/>
            lo a g d:</s>
            <s xml:id="echoid-s11594" xml:space="preserve"> dico, quòd angulus a q b nõ erit minor a g d.</s>
            <s xml:id="echoid-s11595" xml:space="preserve"> Sit enim minor:</s>
            <s xml:id="echoid-s11596" xml:space="preserve"> & ducatur linea g n, diuidẽs an
              <lb/>
            gulũ diametrorũ per ęqualia:</s>
            <s xml:id="echoid-s11597" xml:space="preserve"> [per 9 p 1] & ducatur linea a b, quã diuidat g n per punctũ f.</s>
            <s xml:id="echoid-s11598" xml:space="preserve"> Palàm [per
              <lb/>
            3 p 6] quòd proportio b g ad g a, ſicut b f ad f a:</s>
            <s xml:id="echoid-s11599" xml:space="preserve"> ſed cũ b g maior ſit g a:</s>
            <s xml:id="echoid-s11600" xml:space="preserve"> [ex theſi] erit b f maior f a.</s>
            <s xml:id="echoid-s11601" xml:space="preserve"> Diui
              <lb/>
            datur a b per mediũ in puncto k:</s>
            <s xml:id="echoid-s11602" xml:space="preserve"> [per 10 p 1] & fiat [per 5 p 4] circulus tranſiens per tria puncta a, b, t:</s>
            <s xml:id="echoid-s11603" xml:space="preserve">
              <lb/>
            qui quidẽ circulus nõ tranſibit per g:</s>
            <s xml:id="echoid-s11604" xml:space="preserve"> quoniã anguli a g b, b t a eſſent æquales duobus rectis [per 22 p
              <lb/>
            3] & palàm, quòd ſunt minores:</s>
            <s xml:id="echoid-s11605" xml:space="preserve"> cũ [per theſin] angu
              <lb/>
              <figure xlink:label="fig-0180-01" xlink:href="fig-0180-01a" number="125">
                <variables xml:id="echoid-variables115" xml:space="preserve">z t n q p i b k f e l a n m g h d</variables>
              </figure>
            lus b t a ſit minor angulo a g d [qui cũ angulo a g b ę-
              <lb/>
            quatur duobus rectis per 13 p 1.</s>
            <s xml:id="echoid-s11606" xml:space="preserve">] Igitur trãſibit ſupra
              <lb/>
            g.</s>
            <s xml:id="echoid-s11607" xml:space="preserve"> Similiter nõ trãſibit per q:</s>
            <s xml:id="echoid-s11608" xml:space="preserve"> quoniã ſumpto puncto
              <lb/>
            circuli, in quo linea g q ſecat ipſũ, ſcilicet m:</s>
            <s xml:id="echoid-s11609" xml:space="preserve"> eſſet ar-
              <lb/>
            cus a m æqualis arcui b m [per 26 p 3] cũ reſpiciãt æ-
              <lb/>
            quales angulos ſuper q:</s>
            <s xml:id="echoid-s11610" xml:space="preserve"> [per theſin & 12 n 4:</s>
            <s xml:id="echoid-s11611" xml:space="preserve"> quia q
              <lb/>
            eſt reflexiõis punctũ] quod manet impoſsibile.</s>
            <s xml:id="echoid-s11612" xml:space="preserve"> Quo
              <lb/>
            niam ſumpto puncto o, in quo linea g t ſecat hũc cir.</s>
            <s xml:id="echoid-s11613" xml:space="preserve">
              <lb/>
            culũ:</s>
            <s xml:id="echoid-s11614" xml:space="preserve"> erit arcus a o ęqualis arcui o b:</s>
            <s xml:id="echoid-s11615" xml:space="preserve"> [per 26 p 3] quia
              <lb/>
            reſpiciũt ęquales angulos ſuք t [per theſin & 12 n 4:</s>
            <s xml:id="echoid-s11616" xml:space="preserve">
              <lb/>
            & ſic peripheria b o maior eſſet peripheria b m, pars
              <lb/>
            ſuo toto cõtra 9 ax:</s>
            <s xml:id="echoid-s11617" xml:space="preserve">] Reſtat, ut hic circulus tranſeat
              <lb/>
            ſupra q:</s>
            <s xml:id="echoid-s11618" xml:space="preserve"> ſi enim infra:</s>
            <s xml:id="echoid-s11619" xml:space="preserve"> eadẽ erit improbatio.</s>
            <s xml:id="echoid-s11620" xml:space="preserve"> Ducatur
              <lb/>
            aũt linea à puncto o ad punctũ k:</s>
            <s xml:id="echoid-s11621" xml:space="preserve"> quæ quidẽ cum di-
              <lb/>
            uidat chordã a b per ęqualia:</s>
            <s xml:id="echoid-s11622" xml:space="preserve"> [per fabricationẽ] & ſi-
              <lb/>
            militer arcũ a b:</s>
            <s xml:id="echoid-s11623" xml:space="preserve"> [quia peripheria a o æqualis oſtenſa
              <lb/>
            eſt ipſi b o] erit perpendicularis ſuper a b.</s>
            <s xml:id="echoid-s11624" xml:space="preserve"> [rectæ e-
              <lb/>
            nim lineæ ſubtendentes peripherias a o, b o, æquales ſunt per 29 p 3, & b k æquatur ipſi k a, & cõmu
              <lb/>
            ne latus eſt k o.</s>
            <s xml:id="echoid-s11625" xml:space="preserve"> Quare per 8 p.</s>
            <s xml:id="echoid-s11626" xml:space="preserve"> 10 d 1, o k perpendicularis eſt ipſi a b.</s>
            <s xml:id="echoid-s11627" xml:space="preserve">] Verùm angulus b a g maior an-
              <lb/>
            gulo a b g:</s>
            <s xml:id="echoid-s11628" xml:space="preserve"> [per 18 p 1] cũ b g ſit maior g a:</s>
            <s xml:id="echoid-s11629" xml:space="preserve"> [ex theli] & angulus b f g ualet duos angulos fa g, f g a [per
              <lb/>
            32 p 1] & angulus a f g ualet duos angulos f b g, f g b:</s>
            <s xml:id="echoid-s11630" xml:space="preserve"> ſed a g f ęqualis eſt f g b:</s>
            <s xml:id="echoid-s11631" xml:space="preserve"> [per fabricationẽ] & f a g
              <lb/>
            maior f b g.</s>
            <s xml:id="echoid-s11632" xml:space="preserve"> Igitur angulus b f g maior eſt angulo a f g:</s>
            <s xml:id="echoid-s11633" xml:space="preserve"> igitur b f g maior eſt recto:</s>
            <s xml:id="echoid-s11634" xml:space="preserve"> [per 13 p 1] quare n f
              <lb/>
            b minor eſt recto.</s>
            <s xml:id="echoid-s11635" xml:space="preserve"> [per 13 p 1.</s>
            <s xml:id="echoid-s11636" xml:space="preserve">] Sed o k ſuper f b facit angulũ rectũ:</s>
            <s xml:id="echoid-s11637" xml:space="preserve"> ergo producta cõcurret cũ g n [per
              <lb/>
            11 ax:</s>
            <s xml:id="echoid-s11638" xml:space="preserve">] ſupra b f, & inferius nunꝗ̃.</s>
            <s xml:id="echoid-s11639" xml:space="preserve"> [ſecus per 3 p 6 b k
              <lb/>
              <figure xlink:label="fig-0180-02" xlink:href="fig-0180-02a" number="126">
                <variables xml:id="echoid-variables116" xml:space="preserve">z t n q b k f a e o g h d</variables>
              </figure>
            fieret maior k a, cui eſt æquata.</s>
            <s xml:id="echoid-s11640" xml:space="preserve">] Facto autem circulo
              <lb/>
            trãſeunte per tria pũcta a, q, b:</s>
            <s xml:id="echoid-s11641" xml:space="preserve"> trãſibit ſupra g.</s>
            <s xml:id="echoid-s11642" xml:space="preserve"> [Quia
              <lb/>
            ſi trãſiret per punctũ g:</s>
            <s xml:id="echoid-s11643" xml:space="preserve"> eſſent anguli a q b, a g b æqua
              <lb/>
            les duob.</s>
            <s xml:id="echoid-s11644" xml:space="preserve"> rectis per 22 p 3:</s>
            <s xml:id="echoid-s11645" xml:space="preserve"> & anguli a g b, a g d æquan
              <lb/>
            tur duob.</s>
            <s xml:id="echoid-s11646" xml:space="preserve"> rectis per 13 p 1.</s>
            <s xml:id="echoid-s11647" xml:space="preserve"> Quare per 3 ax.</s>
            <s xml:id="echoid-s11648" xml:space="preserve"> a q b æqua-
              <lb/>
            retur a g d:</s>
            <s xml:id="echoid-s11649" xml:space="preserve"> cõtra præcedentẽ numerũ] & g q diuidet
              <lb/>
            arcũ eius a b per æqualia [quia enim q ex theſi eſt re-
              <lb/>
            flexiõis punctũ:</s>
            <s xml:id="echoid-s11650" xml:space="preserve"> æquãtur anguli g q a, g q b per 12 n 4
              <lb/>
            & per 26 p 3 peripheria a b bifariã ſecabitur à recta g
              <lb/>
            q] ſed k o diuidit chordã a b per æqualia [per fabrica
              <lb/>
            tionẽ.</s>
            <s xml:id="echoid-s11651" xml:space="preserve">] Ergo k o cõcurret cũ g n infra b f, & ſupra pũ-
              <lb/>
            ctũ g.</s>
            <s xml:id="echoid-s11652" xml:space="preserve"> Igitur k o cõcurrens cũ b a, prius cõcurret cum
              <lb/>
            g n infra b f:</s>
            <s xml:id="echoid-s11653" xml:space="preserve"> & iam improbatũ eſt.</s>
            <s xml:id="echoid-s11654" xml:space="preserve"> Reſtat ergo, ut an-
              <lb/>
            gulus a q b nõ ſit minor angulo a g d:</s>
            <s xml:id="echoid-s11655" xml:space="preserve"> aut quòd a nõ
              <lb/>
            reflectetur ad b à pũcto q [cõtra theſin.</s>
            <s xml:id="echoid-s11656" xml:space="preserve">] Similis erit
              <lb/>
            improbatio, ſumpto quolibet puncto arcus e n.</s>
            <s xml:id="echoid-s11657" xml:space="preserve"> Sũ-
              <lb/>
            pto aũt puncto in arcu n z:</s>
            <s xml:id="echoid-s11658" xml:space="preserve"> qui ſit p:</s>
            <s xml:id="echoid-s11659" xml:space="preserve"> fiat reflexio puncti a ad b à puncto p, ut angulus cõſtans ex angu
              <lb/>
            lo incidẽtię & reflexiõis ſuprap, ſit minor angulo a g d, ſicut angulus cõſtãs ex angulo incidẽtię & re
              <lb/>
            flexionis ſuprat, minor eſt eodẽ.</s>
            <s xml:id="echoid-s11660" xml:space="preserve"> Improbabitur aũt hoc modo, Ducãtur a p, b p, g p:</s>
            <s xml:id="echoid-s11661" xml:space="preserve"> oportet ergo ne-
              <lb/>
            </s>
          </p>
        </div>
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