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

Table of figures

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[141] f e t h k o b m a g n d
[142] f e t b m f a g d n
[143] l m a b g n d
[144] e b g q m d a o z h k
[145] a s c p c f d d e b
[146] e b g q l m d o a z n h k
[147] d z b t m l q r p h k f g e a
[148] s z o r x a h k g m u b d e t l f q p n
[149] a b h
[150] a l c q g d b h
[151] a g e u m q d o n z h p l
[152] a e u g d o p h q n k z i s t f
[153] f f e a z b h d g
[154] a f b m k q n e t h d z
[155] b a e p g d
[156] a b h z e p g d
[157] o z l h m n q t d a b e
[158] z i l m h n t d z a k g y c f b z r s u p a e x
[159] i u r c z h t m g b n q f a
[160] i u r k c z l b d t m g n q f a
[161] l u r c z o d t m g b n k q f a s p x e s
[162] d t e h s n q b l q m f p a g
[163] e c h m z b d a
[164] e n c z b d g a
[165] c h z b d g a
[166] b e a d h z m g
[167] p o b c e l m t n a q k f d g
[168] b d a e h t z g f
[169] e b f a d m h t z g
[170] q e a b d m h z
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          <p>
            <s xml:id="echoid-s13411" xml:space="preserve">
              <pb o="195" file="0201" n="201" rhead="OPTICAE LIBER VI."/>
            peripheriæ b m & d f æquabũtur] & ducatur linea a m u:</s>
            <s xml:id="echoid-s13412" xml:space="preserve"> & lineę i b, i g, i m, n m:</s>
            <s xml:id="echoid-s13413" xml:space="preserve"> & linea q m:</s>
            <s xml:id="echoid-s13414" xml:space="preserve"> quę pro-
              <lb/>
            ducatur uſq;</s>
            <s xml:id="echoid-s13415" xml:space="preserve"> ad exteriorẽ circulũ:</s>
            <s xml:id="echoid-s13416" xml:space="preserve"> & cadat in punctũ z:</s>
            <s xml:id="echoid-s13417" xml:space="preserve"> & ducantur lineę z a, z g.</s>
            <s xml:id="echoid-s13418" xml:space="preserve"> Cum aũt arcus b m
              <lb/>
            ſit æqualis arcui d f:</s>
            <s xml:id="echoid-s13419" xml:space="preserve"> addito cõmuni:</s>
            <s xml:id="echoid-s13420" xml:space="preserve"> [m d] erit arcus m f æqualis arcui d b:</s>
            <s xml:id="echoid-s13421" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s13422" xml:space="preserve"> [per 27 p 3] angulus
              <lb/>
            n a m æqualis angulo i a b, & latera lateribus æqualia [per 15 d 1] erit [per 4 p 1] m n æqualis i b:</s>
            <s xml:id="echoid-s13423" xml:space="preserve"> & an-
              <lb/>
            gulus n m a æqualis angulo i b a:</s>
            <s xml:id="echoid-s13424" xml:space="preserve"> & [per 13
              <lb/>
              <figure xlink:label="fig-0201-01" xlink:href="fig-0201-01a" number="159">
                <variables xml:id="echoid-variables149" xml:space="preserve">i u r c z h
                  <gap/>
                t m g b n q f a</variables>
              </figure>
            p 1] angulus n m u angulo i b c.</s>
            <s xml:id="echoid-s13425" xml:space="preserve"> Et cũ poſi
              <lb/>
            ta ſit ſuprà [in ſecunda figura] a q æqualis
              <lb/>
            a h:</s>
            <s xml:id="echoid-s13426" xml:space="preserve"> erunt a q, a m latera æqualia a h, a b:</s>
            <s xml:id="echoid-s13427" xml:space="preserve"> &
              <lb/>
            angulus [q a m] angulo [h a b per proxi-
              <lb/>
            mam cõcluſionẽ] erit [per 4 p 1] q m ęqua
              <lb/>
            lis h b:</s>
            <s xml:id="echoid-s13428" xml:space="preserve"> & erit angulus q m a æqualis h b a,
              <lb/>
            & q m n æqualis angulo h b i:</s>
            <s xml:id="echoid-s13429" xml:space="preserve"> [per 8 p 1]
              <lb/>
            quoniã duo eius latera duobus illius æ-
              <lb/>
            qualia:</s>
            <s xml:id="echoid-s13430" xml:space="preserve"> [nam m n æqualis cõcluſa eſt ipſi
              <lb/>
            i b, & q m ipſi h b] & baſis, quæ eſt q n, eſt
              <lb/>
            æqualis baſi h i:</s>
            <s xml:id="echoid-s13431" xml:space="preserve"> [nam a n, a i æquãtur per
              <lb/>
            15 d 1:</s>
            <s xml:id="echoid-s13432" xml:space="preserve"> itẽ a q, a h per theſin:</s>
            <s xml:id="echoid-s13433" xml:space="preserve"> reliqua igitur
              <lb/>
            q n æquatur reliquæ h i] & angulus n m u
              <lb/>
            æqualis angulo i b c, & i b c æqualis angu
              <lb/>
            lo h b a:</s>
            <s xml:id="echoid-s13434" xml:space="preserve"> [ut oſtenſum eſt in ſecũda figura:</s>
            <s xml:id="echoid-s13435" xml:space="preserve">
              <lb/>
            ubi angulus i b z eſt hic i b c] & angulus h
              <lb/>
            b a ęqualis angulo q m a:</s>
            <s xml:id="echoid-s13436" xml:space="preserve"> ergo n m u ęqua
              <lb/>
            lis q m a.</s>
            <s xml:id="echoid-s13437" xml:space="preserve"> Et quoniam, ut poſuimus, q m z
              <lb/>
            eſt linea recta:</s>
            <s xml:id="echoid-s13438" xml:space="preserve"> erit angulus q m a æqualis angulo u m z [ք 15 p 1:</s>
            <s xml:id="echoid-s13439" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s13440" xml:space="preserve"> anguli n m u, z m u æquãtur.</s>
            <s xml:id="echoid-s13441" xml:space="preserve">
              <lb/>
            Quare punctũn reflectitur ad z à puncto m:</s>
            <s xml:id="echoid-s13442" xml:space="preserve"> [per 12 n 4] & locus imaginis ipſius q [per 3 n 5.</s>
            <s xml:id="echoid-s13443" xml:space="preserve">] Hocta-
              <lb/>
            men deeſt probationi, ut pateat m z totã eſſe extra circulũ:</s>
            <s xml:id="echoid-s13444" xml:space="preserve"> quod ſic patebit.</s>
            <s xml:id="echoid-s13445" xml:space="preserve"> Palàm, quod contingẽs
              <lb/>
            ducta à pũcto b cadat inter i & h:</s>
            <s xml:id="echoid-s13446" xml:space="preserve"> [demõſtratũ enim eſt in prima figura punctũ l alterũ terminũ rectę
              <lb/>
            tangentis peripheriã d b e in puncto b, cadere inter puncta i & h] & tanta eſt remotio puncti b à pun
              <lb/>
            cto h, quãta eſt puncti m à pũcto q[æquales enim cõcluſę ſunt h b, q m] & i h æqualis n q.</s>
            <s xml:id="echoid-s13447" xml:space="preserve"> igitur con-
              <lb/>
            tingẽs, ducta à puncto m cadet inter n & q.</s>
            <s xml:id="echoid-s13448" xml:space="preserve"> Igitur q m ſecat circulũ [quia tangẽte inferior eſt.</s>
            <s xml:id="echoid-s13449" xml:space="preserve">] Qua-
              <lb/>
            re tota m z eſt extra circulũ.</s>
            <s xml:id="echoid-s13450" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13451" xml:space="preserve"> quoniã angulus n m u æ qualis eſt angulo u m z:</s>
            <s xml:id="echoid-s13452" xml:space="preserve"> erit arcus n u
              <lb/>
            æqualis arcui u z.</s>
            <s xml:id="echoid-s13453" xml:space="preserve"> [Quia enim m n ęquatur ipſi m z:</s>
            <s xml:id="echoid-s13454" xml:space="preserve"> cõnexę igitur n u & u z æquãtur per 4 p 1.</s>
            <s xml:id="echoid-s13455" xml:space="preserve"> Quare
              <lb/>
            per 28 p 3 peripherię n u, u z æquãtur] & erit angulus n a u æqualis angulo u a z [per 27 p 3.</s>
            <s xml:id="echoid-s13456" xml:space="preserve">] Sed iam
              <lb/>
            patuit, quòd angulus n a u æqualis eſt angulo i a c:</s>
            <s xml:id="echoid-s13457" xml:space="preserve"> igitur angulus i a c erit æqualis angulo u a z.</s>
            <s xml:id="echoid-s13458" xml:space="preserve"> An-
              <lb/>
            gulus uerò b a g aut erit æqualis angulo g a m:</s>
            <s xml:id="echoid-s13459" xml:space="preserve"> aut minor:</s>
            <s xml:id="echoid-s13460" xml:space="preserve"> aut maior.</s>
            <s xml:id="echoid-s13461" xml:space="preserve"> Sit æqualis.</s>
            <s xml:id="echoid-s13462" xml:space="preserve"> Si igitur ab angulo
              <lb/>
            i a c ſubtrahatur angulus b a g, & ab angulo z a u angulus m a g:</s>
            <s xml:id="echoid-s13463" xml:space="preserve"> remanebit angulus i a g æqualis an-
              <lb/>
            gulo z a g:</s>
            <s xml:id="echoid-s13464" xml:space="preserve"> & erit per 4 p 1 i g æqualis z g, & triangulũ triangulo:</s>
            <s xml:id="echoid-s13465" xml:space="preserve"> & erit angulus i g a æqualis angulo
              <lb/>
            z a g:</s>
            <s xml:id="echoid-s13466" xml:space="preserve"> reſtabit igitur [per 13 p 1] angulus i g r æqualis angulo z g r.</s>
            <s xml:id="echoid-s13467" xml:space="preserve"> Fiat igitur angulo i g r æqualis angu
              <lb/>
            lus t g a:</s>
            <s xml:id="echoid-s13468" xml:space="preserve"> per 23 p 1] erit angulus t g a æqualis angulo z g r.</s>
            <s xml:id="echoid-s13469" xml:space="preserve"> Si igitur t g producatur:</s>
            <s xml:id="echoid-s13470" xml:space="preserve"> ueniet ad z [ք con
              <lb/>
            uerſionẽ 15 p 1 à Proclo ibidẽ demonſtratã.</s>
            <s xml:id="echoid-s13471" xml:space="preserve">] Quare t g z linea recta [per 14 p 1.</s>
            <s xml:id="echoid-s13472" xml:space="preserve">] Igitur i à puncto g re
              <lb/>
            flectitur ad z:</s>
            <s xml:id="echoid-s13473" xml:space="preserve"> & locus imaginis eius eſt punctũ t.</s>
            <s xml:id="echoid-s13474" xml:space="preserve"> Si ergo z ſit uiſus:</s>
            <s xml:id="echoid-s13475" xml:space="preserve"> reflectẽtur ad ipſum duo pũctai,
              <lb/>
            n à duobus punctis m, g:</s>
            <s xml:id="echoid-s13476" xml:space="preserve"> & loca imaginũ puncta t, q.</s>
            <s xml:id="echoid-s13477" xml:space="preserve"> Igitur linea t q erit imago lineæ i n.</s>
            <s xml:id="echoid-s13478" xml:space="preserve"> Probatũ aũt
              <lb/>
            eſt ſuprà, quòd t q æqualis eſt i n.</s>
            <s xml:id="echoid-s13479" xml:space="preserve"> Et ita poteſt accidere in his ſpeculis imaginẽ eſſe æqualẽ rei uiſę.</s>
            <s xml:id="echoid-s13480" xml:space="preserve"> Si
              <lb/>
            uerò angulus b a g fuerit maior angulo
              <lb/>
              <figure xlink:label="fig-0201-02" xlink:href="fig-0201-02a" number="160">
                <variables xml:id="echoid-variables150" xml:space="preserve">i u r k c z l b d t m g n q f a</variables>
              </figure>
            g a m:</s>
            <s xml:id="echoid-s13481" xml:space="preserve"> erit angulus z a g maior àngulo i
              <lb/>
            a g [mutua angulorum ſubductione, ut
              <lb/>
            prius facta.</s>
            <s xml:id="echoid-s13482" xml:space="preserve">] Sit angulus k a g æqualis
              <lb/>
            angulo i a g.</s>
            <s xml:id="echoid-s13483" xml:space="preserve"> Quonia pũctũ k demiſsius
              <lb/>
            puncto z, & punctũ m demiſsius pũcto
              <lb/>
            g:</s>
            <s xml:id="echoid-s13484" xml:space="preserve">linea k g ſecabit lineã z m:</s>
            <s xml:id="echoid-s13485" xml:space="preserve"> ſecet in pũ-
              <lb/>
            ctol.</s>
            <s xml:id="echoid-s13486" xml:space="preserve"> Igitur exiſtẽte uiſu in puncto l, re-
              <lb/>
            flectetur n ad ipſum à pũcto m:</s>
            <s xml:id="echoid-s13487" xml:space="preserve"> & locus
              <lb/>
            imaginis q.</s>
            <s xml:id="echoid-s13488" xml:space="preserve"> Similiter i reflectetur ad i-
              <lb/>
            pſum:</s>
            <s xml:id="echoid-s13489" xml:space="preserve"> & locus imaginis eſt t ſecundum
              <lb/>
            priorẽ probationẽ.</s>
            <s xml:id="echoid-s13490" xml:space="preserve"> Et ita t q imago eſt i
              <lb/>
            n.</s>
            <s xml:id="echoid-s13491" xml:space="preserve"> Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s13492" xml:space="preserve"> Si uerò angulus
              <lb/>
            b a g f uerit minor angulo g a m:</s>
            <s xml:id="echoid-s13493" xml:space="preserve"> erit an
              <lb/>
            gulus z a g minor angulo i a g.</s>
            <s xml:id="echoid-s13494" xml:space="preserve"> Sit angu
              <lb/>
            lus o a g æqualis angulo i a g:</s>
            <s xml:id="echoid-s13495" xml:space="preserve"> & duca-
              <lb/>
            tur linea o g.</s>
            <s xml:id="echoid-s13496" xml:space="preserve"> Palàm, quòd i reflectitur
              <lb/>
            ad o à puncto g.</s>
            <s xml:id="echoid-s13497" xml:space="preserve"> Linea o g aut ſecabit li
              <lb/>
            neam z m q extra circulũ ſpeculi:</s>
            <s xml:id="echoid-s13498" xml:space="preserve"> aut nõ.</s>
            <s xml:id="echoid-s13499" xml:space="preserve"> Si ſecet extra, & uiſus fuerit inpuncto ſectionis:</s>
            <s xml:id="echoid-s13500" xml:space="preserve"> reflectẽtur
              <lb/>
            ad ipſum duo puncta n, i:</s>
            <s xml:id="echoid-s13501" xml:space="preserve"> & loca imaginũ erunt t q.</s>
            <s xml:id="echoid-s13502" xml:space="preserve"> Et ita redit propoſitũ [quod erat imaginẽ æquari
              <lb/>
            uiſibili.</s>
            <s xml:id="echoid-s13503" xml:space="preserve">] Si forſan linea o g ſecet lineã z m q intra circulũ:</s>
            <s xml:id="echoid-s13504" xml:space="preserve"> nõ poterit applicari prędicta probatio.</s>
            <s xml:id="echoid-s13505" xml:space="preserve"> Sed
              <lb/>
            </s>
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