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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[181] t n q z g m b ſ f h r a d e k o
[182] t i y n q g z x m b c ſ f h s r a d p e k o u
[183] f d b g t e h e
[184] e c s ſ o f i g m b k z d t q p h y n r u a x
[185] CIN EMATH EQUE FRANCAISE BIBLIOTHEQUE MUSEE
[186] a e t o f z h g d j c p k b q r
[187] a o u m h z t s n d ſ e q f p
[188] a o u p m h z t x b n y c q s l d g e K f r
[189] f u q b m t n e o z a
[190] f q b u g m c n K p a
[191] d g t K z b e a o ſ h
[192] d g t k n z u e b a o ſ h m r
[193] d g p i t k b e a o l f q h
[194] p d h t z f b g a ſ e k q
[195] t f h a ſ i k d r e z b c m o g
[196] q h f d u o g c r e a n m z b
[197] t f h a p k l i d e z b n r m o g q
[198] ſ m s q c d r b n p t a h e g u i f
[199] q s n p e f o x u m l b z k d h a
[200] k q t ſ n ſ g b o e u z d h a
[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]
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          <p>
            <s xml:id="echoid-s15203" xml:space="preserve">
              <pb o="216" file="0222" n="222" rhead="ALHAZEN"/>
            rauimus [ſuperiore numero] quòd linea f q eſt maior linea m n.</s>
            <s xml:id="echoid-s15204" xml:space="preserve"> T unc quando uiſus fuerit ſuper
              <lb/>
            punctum k, & fuerit linea m n in aliquo uiſibili:</s>
            <s xml:id="echoid-s15205" xml:space="preserve"> tunc uiſus apprehẽdet formam maiorem re uiſa.</s>
            <s xml:id="echoid-s15206" xml:space="preserve"> Et
              <lb/>
            ſic, ſi reuoluerimus totam figuram in circuitu lineæ a u, ipſa immobili:</s>
            <s xml:id="echoid-s15207" xml:space="preserve"> tunc punctum k faciet circu
              <lb/>
            lum perpendicularem ſuper lineam a u.</s>
            <s xml:id="echoid-s15208" xml:space="preserve"> Et ſic omne punctum illius circuli habebit ſitum, reſpectu
              <lb/>
            lineæ comparis m n, ſicut eſt ſitus k reſpectu m n.</s>
            <s xml:id="echoid-s15209" xml:space="preserve"> Si ergo uiſus fuerit in aliquo puncto circumferen
              <lb/>
            tiæ huius circuli, & linea compar lineæ m n, fuerit in ſuperficie alicuius rei uiſæ:</s>
            <s xml:id="echoid-s15210" xml:space="preserve"> tunc uiſus compre
              <lb/>
            hendet formam illius lineæ maiorem.</s>
            <s xml:id="echoid-s15211" xml:space="preserve"> Et ſimiliter ſi extrahamus t k rectè, & poſuerimus in ipſa ali-
              <lb/>
            quod punctum præter k, & extraxerimus lineas ſemper ab illo puncto, quod eſt quaſi punctum k:</s>
            <s xml:id="echoid-s15212" xml:space="preserve">
              <lb/>
            erit modus eius ſicut modus puncti k.</s>
            <s xml:id="echoid-s15213" xml:space="preserve"> Ex his ergo duabus figuris patet, quòd in ſphæricis ſpeculis
              <lb/>
            con cauis & multa & ex multis ſitibus comprehenduntur maiora.</s>
            <s xml:id="echoid-s15214" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div516" type="section" level="0" n="0">
          <head xml:id="echoid-head457" xml:space="preserve" style="it">41. In ſpeculo ſphærico cauo imago interdum æquatur uiſibili: & quæ inter uiſum & ſpecu-
            <lb/>
          lum, euerſa, quæ pone uiſum, erecta eſt. 48 p 8.</head>
          <p>
            <s xml:id="echoid-s15215" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15216" xml:space="preserve"> ſit ſpeculum ſphæricum a b circa centrum e:</s>
            <s xml:id="echoid-s15217" xml:space="preserve"> & extrahamus ſuperficiem tranſeuntem per
              <lb/>
            e:</s>
            <s xml:id="echoid-s15218" xml:space="preserve"> & faciat circulũ a b:</s>
            <s xml:id="echoid-s15219" xml:space="preserve"> & extrahamus ex e lineã e z, quocunq;</s>
            <s xml:id="echoid-s15220" xml:space="preserve"> modo fuerit, uſq;</s>
            <s xml:id="echoid-s15221" xml:space="preserve"> ad g:</s>
            <s xml:id="echoid-s15222" xml:space="preserve"> & ex g extra
              <lb/>
            hamus g d perpendicularem ſuper ſuperficiem circuli a b:</s>
            <s xml:id="echoid-s15223" xml:space="preserve"> [per 12 p 11] & in ipſa ſignemus pun-
              <lb/>
            ctum d, quocunq;</s>
            <s xml:id="echoid-s15224" xml:space="preserve"> modo fuerit:</s>
            <s xml:id="echoid-s15225" xml:space="preserve"> & continuemus d e:</s>
            <s xml:id="echoid-s15226" xml:space="preserve"> & extrahamus ipſam uſq;</s>
            <s xml:id="echoid-s15227" xml:space="preserve"> ad o:</s>
            <s xml:id="echoid-s15228" xml:space="preserve"> & extrahamus
              <lb/>
            e b ita, ut contineat cum e d angulum obtuſum:</s>
            <s xml:id="echoid-s15229" xml:space="preserve"> & extrahamus e a ita, ut contineat cum e d angulũ,
              <lb/>
            æqualem angulo d e b:</s>
            <s xml:id="echoid-s15230" xml:space="preserve"> & continuemus d a, d b.</s>
            <s xml:id="echoid-s15231" xml:space="preserve"> Sic ergo ſuperficies duorum triangulorũ d a e, d b e
              <lb/>
            ſecant ſe ſuper lineam d e:</s>
            <s xml:id="echoid-s15232" xml:space="preserve"> & duo anguli acuti d b e, d a e erunt æquales.</s>
            <s xml:id="echoid-s15233" xml:space="preserve"> [per 4 p 1:</s>
            <s xml:id="echoid-s15234" xml:space="preserve"> nam ſemidiame-
              <lb/>
            tri e a, e b æquantur per 15 d 1, & d e communis eſt:</s>
            <s xml:id="echoid-s15235" xml:space="preserve"> anguliq́;</s>
            <s xml:id="echoid-s15236" xml:space="preserve"> d e a, d e b æquantur per fabricationẽ.</s>
            <s xml:id="echoid-s15237" xml:space="preserve">]
              <lb/>
            Extrahamus ergo ex b lineam in ſuperficie trianguli d e b, continentem cum e b angulum, æqualẽ
              <lb/>
            angulo d b e.</s>
            <s xml:id="echoid-s15238" xml:space="preserve"> Hæc ergo linea cõcurret cum linea d e:</s>
            <s xml:id="echoid-s15239" xml:space="preserve"> quia angulus b e d eſt obtuſus, & angulus, qui
              <lb/>
            eſt apud b, eſt acutus.</s>
            <s xml:id="echoid-s15240" xml:space="preserve"> [quia enim angulus d e b eſt obtuſus per fabricationem, reliquus b e o eſt a-
              <lb/>
            cutus per 13 p 1, & e b o acutus, quia ęquatus eſt d b e acuto.</s>
            <s xml:id="echoid-s15241" xml:space="preserve"> Quare d e, b o cõcurrent per 11 ax.</s>
            <s xml:id="echoid-s15242" xml:space="preserve">] Con
              <lb/>
            currant in o:</s>
            <s xml:id="echoid-s15243" xml:space="preserve"> & extrahamus etiam ex a lineam in ſuperficie trianguli d a e, cõtinentẽ cũ a e angulũ,
              <lb/>
            æqualem angulo d a e.</s>
            <s xml:id="echoid-s15244" xml:space="preserve"> Cõcurret ergo cũ d e in o:</s>
            <s xml:id="echoid-s15245" xml:space="preserve"> quia duo an-
              <lb/>
              <figure xlink:label="fig-0222-01" xlink:href="fig-0222-01a" number="191">
                <variables xml:id="echoid-variables180" xml:space="preserve">d g t K z b e a o ſ h</variables>
              </figure>
            guli a e o, b e o ſunt æquales [per fabricationem & 13 p 1] & an-
              <lb/>
            guli, qui ſunt apud a, b, ſunt æquales [itaq;</s>
            <s xml:id="echoid-s15246" xml:space="preserve"> per 26 p 1 b o, a o æ-
              <lb/>
            quantur:</s>
            <s xml:id="echoid-s15247" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s15248" xml:space="preserve"> concurruntin eodem puncto cõtinuatæ lineæ
              <lb/>
            d e.</s>
            <s xml:id="echoid-s15249" xml:space="preserve">] Et extrahamus e t ita, ut cõtineat cum e b angulum rectũ:</s>
            <s xml:id="echoid-s15250" xml:space="preserve">
              <lb/>
            & extrahamus t e ex parte e, & b o ex parte o:</s>
            <s xml:id="echoid-s15251" xml:space="preserve"> & concurrant in
              <lb/>
            h, [concurrent autẽ per 11 ax:</s>
            <s xml:id="echoid-s15252" xml:space="preserve"> quia angulus h e b rectus eſt per
              <lb/>
            fabricationem, & e b o acutus per concluſionem] & erit e t æ-
              <lb/>
            qualis e h [per 26 p 1:</s>
            <s xml:id="echoid-s15253" xml:space="preserve"> anguli enim ad e deinceps recti æquan-
              <lb/>
            tur:</s>
            <s xml:id="echoid-s15254" xml:space="preserve"> item q́;</s>
            <s xml:id="echoid-s15255" xml:space="preserve"> ad b per fabricationem:</s>
            <s xml:id="echoid-s15256" xml:space="preserve"> & b e commune latus eſt u-
              <lb/>
            triuſq;</s>
            <s xml:id="echoid-s15257" xml:space="preserve"> triangulι b e t, b e h] & b t æqualis b h.</s>
            <s xml:id="echoid-s15258" xml:space="preserve"> Et ſimiliter extra
              <lb/>
            hamus e k ita, ut contineat cum e a angulum rectum:</s>
            <s xml:id="echoid-s15259" xml:space="preserve"> & extra-
              <lb/>
            hamus illã ex parte e:</s>
            <s xml:id="echoid-s15260" xml:space="preserve"> & extrahamus a o, & concurrant in l [con
              <lb/>
            current autem per 11 ax.</s>
            <s xml:id="echoid-s15261" xml:space="preserve"> ut proximè oſtenſum eſt.</s>
            <s xml:id="echoid-s15262" xml:space="preserve">] Sic ergo k e
              <lb/>
            erit æqualis e l, & k a æqualis a l, & t e æqualis e h [per 26 p 1, ut
              <lb/>
            patuit.</s>
            <s xml:id="echoid-s15263" xml:space="preserve">] Et continuemus t k, l h.</s>
            <s xml:id="echoid-s15264" xml:space="preserve"> Erũt ergo æquales [duo enim
              <lb/>
            latera e l, e h æqualia concluſa ſunt duobus lateribus e k, e t, &
              <lb/>
            angulus l e h æquatur angulo k e t per 15 p 1.</s>
            <s xml:id="echoid-s15265" xml:space="preserve"> Quare per 4 p 1 l h,
              <lb/>
            k t æquantur.</s>
            <s xml:id="echoid-s15266" xml:space="preserve">] Si ergo uiſus fuerit in d, & l h fuerit in aliquo ui
              <lb/>
            ſibili:</s>
            <s xml:id="echoid-s15267" xml:space="preserve"> tunc d comprehendet l h in ſpeculo a b:</s>
            <s xml:id="echoid-s15268" xml:space="preserve"> & erit t imago h:</s>
            <s xml:id="echoid-s15269" xml:space="preserve">
              <lb/>
            & k imago l [per 6 n 5.</s>
            <s xml:id="echoid-s15270" xml:space="preserve">] Sic igitur t k erit diameter imaginis l h:</s>
            <s xml:id="echoid-s15271" xml:space="preserve">
              <lb/>
            & eſt ei ęqualis.</s>
            <s xml:id="echoid-s15272" xml:space="preserve"> Si ergo reuoluerimus totam figuram, l h immo
              <lb/>
            bili:</s>
            <s xml:id="echoid-s15273" xml:space="preserve"> tunc d faciet circulum.</s>
            <s xml:id="echoid-s15274" xml:space="preserve"> Et ſi uiſus fuerit in aliquo puncto il
              <lb/>
            lius circumferentiæ, poterit comprehendere aliquod uiſibile,
              <lb/>
            comparlineę l h:</s>
            <s xml:id="echoid-s15275" xml:space="preserve"> & erit imago eius æqualis ei.</s>
            <s xml:id="echoid-s15276" xml:space="preserve"> Et ſimiliter ſi ui-
              <lb/>
            ſus fuerit in o, & res uiſa fuerit t k:</s>
            <s xml:id="echoid-s15277" xml:space="preserve"> erit imago æqualis rei uiſæ.</s>
            <s xml:id="echoid-s15278" xml:space="preserve">
              <lb/>
            Sed tamen cum res uiſa fuerit l h, & uiſus fuerit d, fueritq́;</s>
            <s xml:id="echoid-s15279" xml:space="preserve"> imago t k:</s>
            <s xml:id="echoid-s15280" xml:space="preserve"> erit imago conuerſa:</s>
            <s xml:id="echoid-s15281" xml:space="preserve"> ſi h fuerit
              <lb/>
            in dextra, erit t in ſiniſtra:</s>
            <s xml:id="echoid-s15282" xml:space="preserve"> & ſi h fuerit in ſiniſtra, erit t in dextra:</s>
            <s xml:id="echoid-s15283" xml:space="preserve"> & ſi h fuerit ſupra lineam, erit t infra
              <lb/>
            lineam:</s>
            <s xml:id="echoid-s15284" xml:space="preserve"> & ſimiliter l.</s>
            <s xml:id="echoid-s15285" xml:space="preserve"> Et ſi res uiſa fuerit t k, & uiſus fuerit o, & imago fuerit l h:</s>
            <s xml:id="echoid-s15286" xml:space="preserve"> forma eſt recta.</s>
            <s xml:id="echoid-s15287" xml:space="preserve"> Nam
              <lb/>
            imago l h erit retro uiſum, & comprehendetur ante rem uiſam, ſicut declarauimus in capitulo im a
              <lb/>
            ginis quinti tractatus [60 n.</s>
            <s xml:id="echoid-s15288" xml:space="preserve">] Et uiſus comprehendet h, quod eſt imago t in linea h o, & l, quod eſt
              <lb/>
            imago k, in l o.</s>
            <s xml:id="echoid-s15289" xml:space="preserve"> Patet ergo, quòd in ſpeculis concauis cõprehendatur res uiſa quãdoq;</s>
            <s xml:id="echoid-s15290" xml:space="preserve"> æqualis ſibi.</s>
            <s xml:id="echoid-s15291" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div518" type="section" level="0" n="0">
          <head xml:id="echoid-head458" xml:space="preserve" style="it">42. In ſpeculo ſphærico cauo imago inter uiſum & ſpeculum aliquando minor eſt uiſibili &
            <lb/>
          euerſa: pone uiſum aliquando maior eſt, & erecta. 49 p 8.</head>
          <p>
            <s xml:id="echoid-s15292" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15293" xml:space="preserve"> extrahamus b h rectè:</s>
            <s xml:id="echoid-s15294" xml:space="preserve"> & in ipſa ſignemus r, & cõtinemus r e.</s>
            <s xml:id="echoid-s15295" xml:space="preserve"> Sic ergo angulus r e b erit ob-
              <lb/>
            tuſus:</s>
            <s xml:id="echoid-s15296" xml:space="preserve"> [quia h e b rectus eſt per fabricationẽ] & extrahamus r e ad n.</s>
            <s xml:id="echoid-s15297" xml:space="preserve"> Sic ergo t b erit maior b n:</s>
            <s xml:id="echoid-s15298" xml:space="preserve">
              <lb/>
            [Quia enim angulus b e r obtuſus eſt:</s>
            <s xml:id="echoid-s15299" xml:space="preserve"> ergo r e continuata ultra e faciet cum e b angulum acutũ
              <lb/>
            </s>
          </p>
        </div>
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