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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[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
[231] e a g e z b
[232] k o g e c n a d z f h m l p b
[233] e o k a c n g d z h m l p b
[234] a k r q c n g h l m d p z b
[235] ad m g p h l k q bn z c
[236] a d e i f p m h l k b z q o c
[237] a p k d m e l o g h b z c
[238] a q p k d m e g l o b z f c
[239] a d p m h e ſ g o k b n z c
[240] a h m g e n k z b c ſ d
[241] a h g m x e n k z l b c d
[242] a h g f m r e n k b p q d c ſ
[243] a f h m g e n k b p q d c l
[244] a h m g e r o n k b s z c l d
[245] a b g p e d z m h o h l c
[246] k q f b o r c l m e z f g
[247] b g f t n d h k z a m e
[248] b d g q h n k z o a p e m
[249] g a e h c d b z
[250] d a k g e c b z h
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          <pb o="218" file="0224" n="224" rhead="ALHAZEN"/>
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        <div xml:id="echoid-div522" type="section" level="0" n="0">
          <head xml:id="echoid-head460" xml:space="preserve" style="it">44. Si uiſ{us} ſit citra centrum ſpeculi ſphærici caui, uiſibile ultra: imago tum uiſibilis, tum ui-
            <lb/>
          dentis, euerſa & minor uidebitur. 51 p 8.</head>
          <p>
            <s xml:id="echoid-s15389" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15390" xml:space="preserve"> ſit ſpeculum concauum a b:</s>
            <s xml:id="echoid-s15391" xml:space="preserve"> & centrũ g:</s>
            <s xml:id="echoid-s15392" xml:space="preserve"> & habeat ſuperficiem planam, tranſeuntem per cẽ
              <lb/>
            trum, & faciat circulum a b:</s>
            <s xml:id="echoid-s15393" xml:space="preserve"> & extrahamus lineam g d, quocunque modo ſit:</s>
            <s xml:id="echoid-s15394" xml:space="preserve"> & tranſeat ex parte
              <lb/>
            gad e:</s>
            <s xml:id="echoid-s15395" xml:space="preserve"> & ſit uiſus in e:</s>
            <s xml:id="echoid-s15396" xml:space="preserve"> & ſit t in ſuperficie uiſus:</s>
            <s xml:id="echoid-s15397" xml:space="preserve"> & extrahamus t h perpendiculariter ſuper lineã
              <lb/>
            e d:</s>
            <s xml:id="echoid-s15398" xml:space="preserve"> [per 11 p 1] & ſit z t ęqualis t h:</s>
            <s xml:id="echoid-s15399" xml:space="preserve"> & comprehendat e punctum h ex a:</s>
            <s xml:id="echoid-s15400" xml:space="preserve"> & g h producta in p, compre-
              <lb/>
            hendat arcum a p maiorem quarta circuli.</s>
            <s xml:id="echoid-s15401" xml:space="preserve"> Sic ergo erunt duo puncta a, h, à duobus lateribus puncti
              <lb/>
            g.</s>
            <s xml:id="echoid-s15402" xml:space="preserve"> Nam ſi in eodem eſſent:</s>
            <s xml:id="echoid-s15403" xml:space="preserve"> tunc linea, quæ exiret à ſpeculo ad a, non diuideret angulum, quem conti
              <lb/>
            nent duæ lineę radiales, per ęqualia [ſicq́;</s>
            <s xml:id="echoid-s15404" xml:space="preserve">, ut oſtenſum eſt 66 n 5, reflexio nulla fieret.</s>
            <s xml:id="echoid-s15405" xml:space="preserve">] Et extraha-
              <lb/>
            mus lineas e a, a h, g a, g h:</s>
            <s xml:id="echoid-s15406" xml:space="preserve"> & tranſeat g h rectè ad k:</s>
            <s xml:id="echoid-s15407" xml:space="preserve"> duo ergo anguli apud a erunt ęquales:</s>
            <s xml:id="echoid-s15408" xml:space="preserve"> [per the-
              <lb/>
            ſin & 12 n 4] & erit k imago h [per 6 n 5.</s>
            <s xml:id="echoid-s15409" xml:space="preserve">] Et ſit arcus b d ęqualis arcui d a:</s>
            <s xml:id="echoid-s15410" xml:space="preserve"> [fiet autem ęqualis per 33
              <lb/>
            p 6, ſi per 23 p 1 ęquaueris angulum d g b angulo d g a] & continuemus lineas e b, b z, b g:</s>
            <s xml:id="echoid-s15411" xml:space="preserve"> & extra-
              <lb/>
            hamus z g ad l:</s>
            <s xml:id="echoid-s15412" xml:space="preserve"> & ſecet z b diametrum d g in f.</s>
            <s xml:id="echoid-s15413" xml:space="preserve"> Erunt ergo duo anguli apud b ęquales:</s>
            <s xml:id="echoid-s15414" xml:space="preserve"> [Quia enim
              <lb/>
            a g, b g ęquantur per 15 d 1, & communis eſt g f, angulusq́ue a g f ęquatus eſt angulo b g f:</s>
            <s xml:id="echoid-s15415" xml:space="preserve"> ęquabi-
              <lb/>
            tur baſis a f, baſi b f, & angulus f a g angulo f b g per 4 p 1.</s>
            <s xml:id="echoid-s15416" xml:space="preserve"> Eadem de cauſſa e a g, e b g æquãtur, quia
              <lb/>
            angulus b g e ęquatur angulo a g e per 13 p 1.</s>
            <s xml:id="echoid-s15417" xml:space="preserve"> Quare cum anguli ad a ęquentur, anguli ad b ęquabun-
              <lb/>
            tur.</s>
            <s xml:id="echoid-s15418" xml:space="preserve">] & comprehendetur z à uiſu ex b:</s>
            <s xml:id="echoid-s15419" xml:space="preserve"> [per 12 n 4] & erit punctum l imago z:</s>
            <s xml:id="echoid-s15420" xml:space="preserve"> [per 6 n 5] & cõtinue
              <lb/>
            mus k l:</s>
            <s xml:id="echoid-s15421" xml:space="preserve"> erit ergo k l diameter imaginis z h.</s>
            <s xml:id="echoid-s15422" xml:space="preserve"> Et quia t h eſt perpendicularis ſuper d e, & z t eſt ę-
              <lb/>
            qualis t h:</s>
            <s xml:id="echoid-s15423" xml:space="preserve"> erunt duę lineę e a, a h ęquales duabus li-
              <lb/>
              <figure xlink:label="fig-0224-01" xlink:href="fig-0224-01a" number="194">
                <variables xml:id="echoid-variables183" xml:space="preserve">p d h t z f b g a ſ
                  <unsure/>
                e k q</variables>
              </figure>
            neis e b, b z:</s>
            <s xml:id="echoid-s15424" xml:space="preserve"> [Quia enim t h, z t ęquantur per fabrica
              <lb/>
            tionem, & t f communis, anguliq́ue ad t recti ſunt:</s>
            <s xml:id="echoid-s15425" xml:space="preserve"> ę-
              <lb/>
            quabitur baſis h f baſi z f per 4 p 1:</s>
            <s xml:id="echoid-s15426" xml:space="preserve"> & a fiam antè ę-
              <lb/>
            qualis concluſa eſt ipſi b f:</s>
            <s xml:id="echoid-s15427" xml:space="preserve"> itaque tota a h ęquatur to
              <lb/>
            ti b z, & a e, b e ęquantur è cõcluſo] & duo anguli a-
              <lb/>
            pud a ſunt ęquales duobus angulis apud b:</s>
            <s xml:id="echoid-s15428" xml:space="preserve"> erit h e
              <lb/>
            ęqualis z e:</s>
            <s xml:id="echoid-s15429" xml:space="preserve"> [per 4 p 1] & linea g h eſt ęqualis lineę
              <lb/>
            z h [per 4 p 1:</s>
            <s xml:id="echoid-s15430" xml:space="preserve"> quia z t, t h ęquantur per fabricatio-
              <lb/>
            nem, & communis eſt t g, anguliq́ue ad t recti ſunt.</s>
            <s xml:id="echoid-s15431" xml:space="preserve">]
              <lb/>
            Ergo duæ lineę a g, g h ſunt ęquales duabus lineis
              <lb/>
            b g, g z, & baſis a h eſt ęqualis baſi b z:</s>
            <s xml:id="echoid-s15432" xml:space="preserve"> ergo [per 8 p
              <lb/>
            1] angulus a h k eſt ęqualis angulo b z l, & angulus h
              <lb/>
            a k eſt ęqualis z b l:</s>
            <s xml:id="echoid-s15433" xml:space="preserve"> ergo h k eſt ęqualis z l [per 26 p 1:</s>
            <s xml:id="echoid-s15434" xml:space="preserve">
              <lb/>
            quia z b ęqualis concluſa eſt ipſi h a] & linea h g eſt
              <lb/>
            ęqualis z g:</s>
            <s xml:id="echoid-s15435" xml:space="preserve"> [è concluſo] ergo g k eſt æqualis g l:</s>
            <s xml:id="echoid-s15436" xml:space="preserve"> [per
              <lb/>
            19 p 5] ergo k l eſt ęquidiſtans z h, [per 27 p 1:</s>
            <s xml:id="echoid-s15437" xml:space="preserve"> nam
              <lb/>
            cum anguli ad uerticem g ęquentur per 15 p 1:</s>
            <s xml:id="echoid-s15438" xml:space="preserve"> ſitq́ue
              <lb/>
            per 7 p 5 l g ad g k, ſicut g z ad g h:</s>
            <s xml:id="echoid-s15439" xml:space="preserve"> ęquabitur per 6 p 6 angulus z l k angulo l z h.</s>
            <s xml:id="echoid-s15440" xml:space="preserve">] Item angulus h g a
              <lb/>
            eſt obtuſus [ex theſi & 33 p 6] & duo anguli apud a ſunt æquales:</s>
            <s xml:id="echoid-s15441" xml:space="preserve"> ergo linea g h eſt maior linea g k:</s>
            <s xml:id="echoid-s15442" xml:space="preserve">
              <lb/>
            [Nam quia angulus a g h obtuſus:</s>
            <s xml:id="echoid-s15443" xml:space="preserve"> erit per 13 p 1 angulus a g k acutus, & h a g, g a k ſunt ęquales ex
              <lb/>
            theſi
              <unsure/>
            :</s>
            <s xml:id="echoid-s15444" xml:space="preserve"> quia punctum a eſt punctum reflexionis:</s>
            <s xml:id="echoid-s15445" xml:space="preserve"> quare per 32 p 1 angulus a k g maior eſt angulo a h k:</s>
            <s xml:id="echoid-s15446" xml:space="preserve">
              <lb/>
            & per 19 p 1 in triangulo a h k latus a h maius eſt a k:</s>
            <s xml:id="echoid-s15447" xml:space="preserve"> ſed ut a h ad a k, ſic h g ad g k per 3 p 6:</s>
            <s xml:id="echoid-s15448" xml:space="preserve"> quia angu
              <lb/>
            li ad a æquales.</s>
            <s xml:id="echoid-s15449" xml:space="preserve"> Itaque cum a h maior ſit a k:</s>
            <s xml:id="echoid-s15450" xml:space="preserve"> erit h g maior g k] & ſimiliter z g eſt maior, quàm g l.</s>
            <s xml:id="echoid-s15451" xml:space="preserve"> Li-
              <lb/>
            nea ergo k l eſt minor, quàm z h [cum enim triangula k g l, h g z ſint ęquiangula per 15 p 1.</s>
            <s xml:id="echoid-s15452" xml:space="preserve"> 6 p 6:</s>
            <s xml:id="echoid-s15453" xml:space="preserve"> erit ք
              <lb/>
            4 p 6, ut g k ad g h, ſic k l ad z h:</s>
            <s xml:id="echoid-s15454" xml:space="preserve"> & cum g k ſit minor g h, erit k l minor z h.</s>
            <s xml:id="echoid-s15455" xml:space="preserve">] Sed k l eſt diameterima-
              <lb/>
            ginis z h:</s>
            <s xml:id="echoid-s15456" xml:space="preserve"> ergo z h uidetur minor, quàm ſit ſecundum ueritatem:</s>
            <s xml:id="echoid-s15457" xml:space="preserve"> & linea z h eſt ſuperficies faciei a-
              <lb/>
              <gap/>
            picientis.</s>
            <s xml:id="echoid-s15458" xml:space="preserve"> Si ergo reuoluerimus circulum a d b, e d immobili:</s>
            <s xml:id="echoid-s15459" xml:space="preserve"> fiet ex duobus punctis a, b circulus
              <lb/>
            in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s15460" xml:space="preserve"> & erit ſitus uiſus e, reſpectu cuiuslibet comparis lineæ z h ex illo circulo, quem
              <lb/>
            ſignant puncta z, h, & ex omni arcu compari arcui a b ex portione ſpeculi, quam diuidit circulus,
              <lb/>
            quem ſignant duo puncta a, b, ſicut eſt ſitus, quem uiſus e habet ex linea z h, & ex arcu a b.</s>
            <s xml:id="echoid-s15461" xml:space="preserve"> Et ſimili-
              <lb/>
            ter declarabitur, ſi poſuerimus lineã z h maiorem, aut minorẽ.</s>
            <s xml:id="echoid-s15462" xml:space="preserve"> Patet ergo ex his omnib.</s>
            <s xml:id="echoid-s15463" xml:space="preserve"> quòd diame
              <lb/>
            ter ſuperficiei faciei aſpicientis cõprehenditur in ſpeculo cõcauo minor, ꝗ̃ ſit.</s>
            <s xml:id="echoid-s15464" xml:space="preserve"> Sciendum ergo, quòd
              <lb/>
            ſi fuerit uiſus in e:</s>
            <s xml:id="echoid-s15465" xml:space="preserve"> tunc aſpiciens comprehẽdet formam ſuam minorem, ꝗ̃ ſit.</s>
            <s xml:id="echoid-s15466" xml:space="preserve"> Et quia k eſt imago h,
              <lb/>
            & l eſt imago z:</s>
            <s xml:id="echoid-s15467" xml:space="preserve"> erit imago cõuerſa.</s>
            <s xml:id="echoid-s15468" xml:space="preserve"> Et ſic uifus e cõprehendet ſuam formam ſecundum quod eſt de-
              <lb/>
            xtrũ in ſiniſtro, & ſurſum deorſum, & è contrario.</s>
            <s xml:id="echoid-s15469" xml:space="preserve"> Similiter ſi uiſus fuerit in quolibet puncto, inter
              <lb/>
            quod & ſuperficiẽ ſpeculi fuerit centrũ ſpeculi:</s>
            <s xml:id="echoid-s15470" xml:space="preserve"> cõprehendet formã ſuã conuerſam.</s>
            <s xml:id="echoid-s15471" xml:space="preserve"> Et hoc eſt quod
              <lb/>
            uoluimus.</s>
            <s xml:id="echoid-s15472" xml:space="preserve"> Patet ergo ex his quatuor figuris, quòd in ſpeculo concauo imago quandoq;</s>
            <s xml:id="echoid-s15473" xml:space="preserve"> comprehẽ-
              <lb/>
            ditur maior:</s>
            <s xml:id="echoid-s15474" xml:space="preserve"> quandoq;</s>
            <s xml:id="echoid-s15475" xml:space="preserve"> minor:</s>
            <s xml:id="echoid-s15476" xml:space="preserve"> quandoq;</s>
            <s xml:id="echoid-s15477" xml:space="preserve"> ęqualis:</s>
            <s xml:id="echoid-s15478" xml:space="preserve"> & nunc recta, nunc conuerſa.</s>
            <s xml:id="echoid-s15479" xml:space="preserve"> Et in capitulo de ima
              <lb/>
            gine [72 n 5] diximus, quòd in ſpeculo cõcauo imago quandoq;</s>
            <s xml:id="echoid-s15480" xml:space="preserve"> erit una:</s>
            <s xml:id="echoid-s15481" xml:space="preserve"> quandoq;</s>
            <s xml:id="echoid-s15482" xml:space="preserve"> duę:</s>
            <s xml:id="echoid-s15483" xml:space="preserve"> quandoq;</s>
            <s xml:id="echoid-s15484" xml:space="preserve">
              <lb/>
            tres:</s>
            <s xml:id="echoid-s15485" xml:space="preserve"> & quandoq;</s>
            <s xml:id="echoid-s15486" xml:space="preserve"> quatuor:</s>
            <s xml:id="echoid-s15487" xml:space="preserve"> & hoc idem accidit in his prędictis.</s>
            <s xml:id="echoid-s15488" xml:space="preserve"> Illud ergo, quod habet imaginem ſe
              <lb/>
            maiorem, fortè habebit alias minores & ęquales:</s>
            <s xml:id="echoid-s15489" xml:space="preserve"> & quod imaginem habet minorem, fortè habebit
              <lb/>
            alias maiores & minores.</s>
            <s xml:id="echoid-s15490" xml:space="preserve"> Et quod rectum uidebitur, fortè uidebitur ſub alia imagine conuerſum, &
              <lb/>
            è contrario.</s>
            <s xml:id="echoid-s15491" xml:space="preserve"> Reſtat ergo declarare formas eorum, quæ comprehenduntur in his ſpeculis.</s>
            <s xml:id="echoid-s15492" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div524" type="section" level="0" n="0">
          <head xml:id="echoid-head461" xml:space="preserve" style="it">45. In ſpeculo ſphærico cauo imago lineæ rectæ aliquando uidetur recta. Et ſiduo lineæ rectæ
            <lb/>
          termini reflectantur à duob{us} punctis peripheriæ circuli (qui eſt communis ſectio ſuperficie-
            <lb/>
          </head>
        </div>
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