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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[201] k q p t ſ n g b o r f e u m z d h a
[202] t i n g y z x q m b c œ f h z r a d p e K o
[203] u r h d x b y m ſ o n f g i k q z t c c s a
[204] p b o n m d r h c t a K
[205] d g p i t k n u b e a o f q l h m r
[206] a h p u m z t x b n c q s d g ſ K f r
[207] d g p i t k n z u b e a ſ o q l h m r
[208] h n m ſ a s x t r c e d z b g o p q k
[209] u g z y x r s t
[Figure 210]
[Figure 211]
[212] a h e d c b k q l g f
[213] a d c g b e f
[214] k n m x b l p f s u z y t
[215] k n b l o q f g u z
[216] k n m b l d p o q f g u
[217] k b d o f u g z r e a
[218] k h b m z d e a t i g
[219] h m k o n q e f p g i
[220] a k h g p d b c l
[221] a p h f l g e o k a n m e z q b
[222] a f h p g o e k d m n c q z b
[223] a f h p l g o e k d b m c q z n
[224] a f l p g e o k d b n m c z
[225] h a b g e f d e z
[226] h a b e d c z
[227] e a b d f c
[228] a r c p e h b z b d
[229] a n r l c x m h e p z g b b f d o k
[230] a l g h e z d k b t
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          <p>
            <s xml:id="echoid-s16175" xml:space="preserve">
              <pb o="227" file="0233" n="233" rhead="OPTICAE LIBER VI."/>
            bus punctis h, t:</s>
            <s xml:id="echoid-s16176" xml:space="preserve"> & media earum erunt diſtincta & ſeparata:</s>
            <s xml:id="echoid-s16177" xml:space="preserve"> & h t eſt diameter imaginis s i, quocun-
              <lb/>
            que modo fuerit imago:</s>
            <s xml:id="echoid-s16178" xml:space="preserve"> & diameter eſt cõmunis omnibus imaginibus eius, ſi plures habuerit ima-
              <lb/>
            gines:</s>
            <s xml:id="echoid-s16179" xml:space="preserve"> & linea h t eſt maior, quàm si, modica quantitate.</s>
            <s xml:id="echoid-s16180" xml:space="preserve"> Patet ergo, quòd cum lineæ rectæ, æquidi-
              <lb/>
            ſtantes axi columnaris ſpeculi concaui fuerint in aliquo uiſibili:</s>
            <s xml:id="echoid-s16181" xml:space="preserve"> imago earum fortè erit recta aut
              <lb/>
            concaua, & fortè una, aut plures.</s>
            <s xml:id="echoid-s16182" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div539" type="section" level="0" n="0">
          <head xml:id="echoid-head469" xml:space="preserve" style="it">52. Si uiſ{us} à terminis lineæ rectæ æquabiliter diſtans, ſit extra ipſi{us} planum, perpendicula
            <lb/>
          re plano axis ſpeculi cylindr acei caui: imago uidebitur maximè caua. 27 p 9.</head>
          <p>
            <s xml:id="echoid-s16183" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s16184" xml:space="preserve"> iteremus ſecundam figuram de fallacijs ſpeculorum columnarium conuexorum [29 n.</s>
            <s xml:id="echoid-s16185" xml:space="preserve">]
              <lb/>
            In hac autem figura dictum eſt:</s>
            <s xml:id="echoid-s16186" xml:space="preserve"> quòd duæ lineæ e b, h b reflectuntur ſecũdum angulos æquales:</s>
            <s xml:id="echoid-s16187" xml:space="preserve">
              <lb/>
            & quòd duæ lineæ e g, t g reflectuntur ſecundum angulos æquales:</s>
            <s xml:id="echoid-s16188" xml:space="preserve"> & quòd h b, t g perueniunt
              <lb/>
            a d l:</s>
            <s xml:id="echoid-s16189" xml:space="preserve"> & h b continet cum b o angulum acutum.</s>
            <s xml:id="echoid-s16190" xml:space="preserve"> Ergo h b ſecat ſuperficiem, contingentem columnam
              <lb/>
            in b:</s>
            <s xml:id="echoid-s16191" xml:space="preserve"> b l ergo eſt ſub concauitate columnæ:</s>
            <s xml:id="echoid-s16192" xml:space="preserve"> & ſimiliter g l:</s>
            <s xml:id="echoid-s16193" xml:space="preserve"> & ſimiliter duæ lineæ b r, g y:</s>
            <s xml:id="echoid-s16194" xml:space="preserve"> & duo angu-
              <lb/>
            li l b d, d b r ſunt æquales [quia per 15 p 1 æquantur angulis e b o, h b o æqualibus] & ſimiliter l g d, g
              <lb/>
            d y ſunt æquales.</s>
            <s xml:id="echoid-s16195" xml:space="preserve"> Si ergo r y fuerit in aliquo uiſibili, & uiſus fuerit in l, & ſuperficies concaua colu-
              <lb/>
            mnæ fuerit terſa:</s>
            <s xml:id="echoid-s16196" xml:space="preserve"> tunc forma r extenditur per r b, & peruenit ad b, & reflectitur ք b l:</s>
            <s xml:id="echoid-s16197" xml:space="preserve"> & perueniet ad
              <lb/>
            l, & comprehendetur in l.</s>
            <s xml:id="echoid-s16198" xml:space="preserve"> Et linea h u eſt perpendicularis ſuper lineam, contingentem ſectionem,
              <lb/>
            ex cuius circumferentia reflectentur duæ lineæ b r, b l:</s>
            <s xml:id="echoid-s16199" xml:space="preserve"> h ergo eſt imago r [per 7 n 5.</s>
            <s xml:id="echoid-s16200" xml:space="preserve">] Similiter decla
              <lb/>
            rabitur, quòd forma y extenditur per y g, & reflectitur ք g l:</s>
            <s xml:id="echoid-s16201" xml:space="preserve"> & imago eius eſt t.</s>
            <s xml:id="echoid-s16202" xml:space="preserve"> Et continuemus q u:</s>
            <s xml:id="echoid-s16203" xml:space="preserve">
              <lb/>
            ſecabit ergo r y in m:</s>
            <s xml:id="echoid-s16204" xml:space="preserve"> m ergo eſt in ſuperficie tranſeunte per axem & per l:</s>
            <s xml:id="echoid-s16205" xml:space="preserve"> nam l & q ſunt in hac ſu-
              <lb/>
            perficie, [ut demonſtratum eſt 29 n.</s>
            <s xml:id="echoid-s16206" xml:space="preserve">] Ergo q u eſt in hac ſuperficie [nam 29 n oſten ſum eſt, quòd
              <lb/>
            planum ductum per uiſum & axem ſpeculi, in quo eſt linea e l d, ſecat lineam h t in puncto q:</s>
            <s xml:id="echoid-s16207" xml:space="preserve"> eſtq́ue
              <lb/>
            punctum u in linea e l d:</s>
            <s xml:id="echoid-s16208" xml:space="preserve"> linea igitur q u eſt in plano per uiſum & axem ſpeculi ducto per 1 p 11:</s>
            <s xml:id="echoid-s16209" xml:space="preserve"> ideõ-
              <lb/>
            que & punctum m.</s>
            <s xml:id="echoid-s16210" xml:space="preserve">] Et quia duo puncta m, l ſunt in ſuperficie tranſeunte per axem columnæ:</s>
            <s xml:id="echoid-s16211" xml:space="preserve"> ideo
              <lb/>
            forma m reflectetur ad l in hac ſuperficie.</s>
            <s xml:id="echoid-s16212" xml:space="preserve"> Et quia a z eſt differentia communis inter columnę ſuper-
              <lb/>
            ficiem, & ſuperficiem, tranſeuntem per ſuum axem, & per l:</s>
            <s xml:id="echoid-s16213" xml:space="preserve"> forma ergo m reflectetur à linea a z.</s>
            <s xml:id="echoid-s16214" xml:space="preserve"> Et
              <lb/>
            continuemus e m, quæ eſt in hac ſuperficie:</s>
            <s xml:id="echoid-s16215" xml:space="preserve"> & e l
              <lb/>
              <figure xlink:label="fig-0233-01" xlink:href="fig-0233-01a" number="203">
                <variables xml:id="echoid-variables192" xml:space="preserve">u r h d x b y m ſ o n f g i k q z t c c s a</variables>
              </figure>
            eſt in hac ſuperficie:</s>
            <s xml:id="echoid-s16216" xml:space="preserve"> & punctum e eſt elongatum à
              <lb/>
            ſuperficie contingente ſuperficiem columnæ in li-
              <lb/>
            nea a z [ut patuit 29 n.</s>
            <s xml:id="echoid-s16217" xml:space="preserve">] Ergo ſi a z extrahatur re-
              <lb/>
            ctè in parte z:</s>
            <s xml:id="echoid-s16218" xml:space="preserve"> concurret cum duabus lineis e m,
              <lb/>
            e l.</s>
            <s xml:id="echoid-s16219" xml:space="preserve"> Concurrat ergo cum e m in i, & cum e l in n:</s>
            <s xml:id="echoid-s16220" xml:space="preserve">
              <lb/>
            ergo n eſt inter duo puncta e, l:</s>
            <s xml:id="echoid-s16221" xml:space="preserve"> quia l eſt intra con
              <lb/>
            cauitatem columnæ, & n eſt in ſuperficie colu-
              <lb/>
            mnæ:</s>
            <s xml:id="echoid-s16222" xml:space="preserve"> & e eſt elõgatum à columna:</s>
            <s xml:id="echoid-s16223" xml:space="preserve"> & in dem on-
              <lb/>
            ſtratione huius figuræ [29 n] patuit, quòd circu-
              <lb/>
            lus b g eſt medius inter lineam h t, & ſuperficiem
              <lb/>
            exeuntem ex e, æ quidiſtantem baſibus columnæ:</s>
            <s xml:id="echoid-s16224" xml:space="preserve">
              <lb/>
            & perpendicularis, quæ exit ex e ſuper a z, eſt in ſu
              <lb/>
            perficie exeunte ex e, æ quidiſtante columnæ.</s>
            <s xml:id="echoid-s16225" xml:space="preserve"> Er-
              <lb/>
            go perpendicularis, quæ exit ex e ſuper lineam a
              <lb/>
            z n, cadit extra triangulum e i n, & in parte n:</s>
            <s xml:id="echoid-s16226" xml:space="preserve"> angu
              <lb/>
            lus ergo e i n eſt acutus:</s>
            <s xml:id="echoid-s16227" xml:space="preserve"> [per 32 p 1] ergo [per 15 p
              <lb/>
            1] angulus m i a eſt acutus:</s>
            <s xml:id="echoid-s16228" xml:space="preserve"> ergo m i n obtuſus [per
              <lb/>
            13 p 1.</s>
            <s xml:id="echoid-s16229" xml:space="preserve">] Extrahamus ergo ex m perpendicularem
              <lb/>
            ſuper a i [per 12 p 1] & ſit m k:</s>
            <s xml:id="echoid-s16230" xml:space="preserve"> k ergo erit ultra i,
              <lb/>
            reſpectu 11.</s>
            <s xml:id="echoid-s16231" xml:space="preserve"> [ſi enim caderet inter i & n:</s>
            <s xml:id="echoid-s16232" xml:space="preserve"> eſſent triã-
              <lb/>
            guli tres anguli maiores duobus rectis contra 32
              <lb/>
            p 1:</s>
            <s xml:id="echoid-s16233" xml:space="preserve"> quia angulus m i n obtuſus eſt concluſus.</s>
            <s xml:id="echoid-s16234" xml:space="preserve">] Et
              <lb/>
            extrahamus m k ex parte k, in s:</s>
            <s xml:id="echoid-s16235" xml:space="preserve"> & diuidamus k s
              <lb/>
            ad æqualitatem k m:</s>
            <s xml:id="echoid-s16236" xml:space="preserve"> ergo s erit extra ſuperficiem
              <lb/>
            ſpeculi, & ultra concauitatem eius, & l erit ſub concauitate eius.</s>
            <s xml:id="echoid-s16237" xml:space="preserve"> Et continuemus l s:</s>
            <s xml:id="echoid-s16238" xml:space="preserve"> ſecabit ergo
              <lb/>
            n k in f:</s>
            <s xml:id="echoid-s16239" xml:space="preserve"> & ex f extrahamus f x ad æquidiſtantiam m k.</s>
            <s xml:id="echoid-s16240" xml:space="preserve"> Cum ergo [per 29 p 1] f x ſit perpendicularis
              <lb/>
            ſuper a n, & in ſuperficie tranſeunte per axem & per l:</s>
            <s xml:id="echoid-s16241" xml:space="preserve"> ergo eſt diameter circuli exeuntis ex f & æ-
              <lb/>
            quidiſtantis baſi columnæ [per 34 n 4.</s>
            <s xml:id="echoid-s16242" xml:space="preserve">] Linea ergo f x eſt perpendicularis ſuper ſuperficiem, con-
              <lb/>
            tingentem columnam, tranſeuntem per a z [ſicut oſtenſum eſt 54 n 5.</s>
            <s xml:id="echoid-s16243" xml:space="preserve">] Et continuemus m f:</s>
            <s xml:id="echoid-s16244" xml:space="preserve"> erit
              <lb/>
            ergo æqualis f s:</s>
            <s xml:id="echoid-s16245" xml:space="preserve"> [per 4 p 1:</s>
            <s xml:id="echoid-s16246" xml:space="preserve"> quia k s, k m æquantur per fabricationem, & communis eſt k f, anguli-
              <lb/>
            que ad k recti] & duo anguli qui ſunt, apud m, s erunt æquales:</s>
            <s xml:id="echoid-s16247" xml:space="preserve"> [per 5 p 1.</s>
            <s xml:id="echoid-s16248" xml:space="preserve">] Et quia x f eſt æquidi-
              <lb/>
            ſtans m g:</s>
            <s xml:id="echoid-s16249" xml:space="preserve"> erunt [per 29 p 1] duo anguli apud f æquales duobus angulis, qui ſunt apud s, m [ideó-
              <lb/>
            que anguli x f m & x f l æquabuntur.</s>
            <s xml:id="echoid-s16250" xml:space="preserve">] Duæ ergo lineę m f, f l reflectuntur ſecundum angu-
              <lb/>
            los æquales:</s>
            <s xml:id="echoid-s16251" xml:space="preserve"> & x f eſt perpendicularis ſuper ſuperficiem, contingentem ſpeculum in f.</s>
            <s xml:id="echoid-s16252" xml:space="preserve"> For-
              <lb/>
            ma ergo m extenditur per m f, & reflectitur per f l:</s>
            <s xml:id="echoid-s16253" xml:space="preserve"> & imago eius erit s [per 7 n 5.</s>
            <s xml:id="echoid-s16254" xml:space="preserve">] Et quia
              <lb/>
            duæ lineæ r y, h t ſunt æquidiſtantes, & perpendiculares ſuper ſuperficiem tranſeuntem per
              <lb/>
            axem, & per l:</s>
            <s xml:id="echoid-s16255" xml:space="preserve"> quia h t fuit poſita talis:</s>
            <s xml:id="echoid-s16256" xml:space="preserve"> [29 n] ideo duæ ſuperficies exeuntes à duabus li-
              <lb/>
            </s>
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