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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[171] l k x s y e t q b a f u r m h o m z g p d
[172] ſ k x b a s t c q f m o h z i g p d
[173] d a b e h z g
[174] d a b e h z g
[175] a d b b g
[176] a d f b ſ m e c z g
[177] h e m c u t s k o b z ſ q r f g a d
[178] h e m c u s t b o q z r f g a d
[179] i h e m c t z u s b o k q r f g a d
[180] n q e ſ g t f m o K d h c a s u p z b
[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
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          <p>
            <s xml:id="echoid-s14641" xml:space="preserve">
              <pb o="210" file="0216" n="216" rhead="ALHAZEN"/>
            hæc perpendicularis concurret cum perpendiculari ſuper contin gentem ſectionem ducta à pun-
              <lb/>
            cto reflexionis, ſub axe.</s>
            <s xml:id="echoid-s14642" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s14643" xml:space="preserve"> ſit a b g z pyramis erecta ſuper baſim ſuam:</s>
            <s xml:id="echoid-s14644" xml:space="preserve"> a uertex pyrami-
              <lb/>
            dis:</s>
            <s xml:id="echoid-s14645" xml:space="preserve"> b f z ſectio:</s>
            <s xml:id="echoid-s14646" xml:space="preserve"> e punctum reflexionis:</s>
            <s xml:id="echoid-s14647" xml:space="preserve"> z punctũ ſectionis remotius à puncto a quàm e.</s>
            <s xml:id="echoid-s14648" xml:space="preserve"> Super pun-
              <lb/>
            ctum z fiat ſuperficies ſecans pyramidem æquidiſtanter baſi [ut oſtenſum eſt 52 n 5.</s>
            <s xml:id="echoid-s14649" xml:space="preserve">] Secabit qui-
              <lb/>
            dem ſuper circulum communem [per 4 th.</s>
            <s xml:id="echoid-s14650" xml:space="preserve"> 1 coni.</s>
            <s xml:id="echoid-s14651" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s14652" xml:space="preserve">] Sit circulus ille g b r z:</s>
            <s xml:id="echoid-s14653" xml:space="preserve"> & ducantur lineæ
              <lb/>
            a z, a e:</s>
            <s xml:id="echoid-s14654" xml:space="preserve"> & producatur a e, donec ſit æqualis a z:</s>
            <s xml:id="echoid-s14655" xml:space="preserve"> ueniet quidem ad circulum [per 18 d 11:</s>
            <s xml:id="echoid-s14656" xml:space="preserve"> quia eſt la-
              <lb/>
            tus conicum.</s>
            <s xml:id="echoid-s14657" xml:space="preserve">] Cadat ergo in punctum eius o:</s>
            <s xml:id="echoid-s14658" xml:space="preserve"> & c ſit centrum circuli:</s>
            <s xml:id="echoid-s14659" xml:space="preserve"> & ducatur axis a c:</s>
            <s xml:id="echoid-s14660" xml:space="preserve"> & à pun-
              <lb/>
            cto e ducatur perpendicularis ſuper ſuperficiem contingentem pyramidem [per 12 p 11.</s>
            <s xml:id="echoid-s14661" xml:space="preserve">] Concur-
              <lb/>
            ret quidem [per 11 ax.</s>
            <s xml:id="echoid-s14662" xml:space="preserve">] cum axe citra cẽtrum circuli, quod eſt c:</s>
            <s xml:id="echoid-s14663" xml:space="preserve"> ſit in puncto d:</s>
            <s xml:id="echoid-s14664" xml:space="preserve"> & ducatur linea d z,
              <lb/>
            continens angulum acutum cum perpendiculari e d:</s>
            <s xml:id="echoid-s14665" xml:space="preserve"> & à puncto o ducatur perpendicularis ſuper
              <lb/>
            lineam a o, concurrens cum axe in puncto k:</s>
            <s xml:id="echoid-s14666" xml:space="preserve"> & ducatur linea k z:</s>
            <s xml:id="echoid-s14667" xml:space="preserve"> & ſuper punctum z ducatur con-
              <lb/>
            tingens ſectionem, quæ ſit t q:</s>
            <s xml:id="echoid-s14668" xml:space="preserve"> & alia contingens circulum b g z:</s>
            <s xml:id="echoid-s14669" xml:space="preserve"> [per 17 p 3] quæ ſit z y:</s>
            <s xml:id="echoid-s14670" xml:space="preserve"> & ducatur
              <lb/>
            linea b c z:</s>
            <s xml:id="echoid-s14671" xml:space="preserve"> & à puncto c ducatur perpendicularis ſuper lineam b c z:</s>
            <s xml:id="echoid-s14672" xml:space="preserve"> [per 11 p 1] quæ ſit c r.</s>
            <s xml:id="echoid-s14673" xml:space="preserve"> Erit qui-
              <lb/>
            dem perpendicularis ſuper axem:</s>
            <s xml:id="echoid-s14674" xml:space="preserve"> [per 3 d 11] cum axis ſit perpendicularis ſuper ſuperficiem circu-
              <lb/>
            li:</s>
            <s xml:id="echoid-s14675" xml:space="preserve"> [per 18 d 11.</s>
            <s xml:id="echoid-s14676" xml:space="preserve">] Quare [per 4 p 11] c r eſt perpendicularis ſuper ſuperficiem a c z:</s>
            <s xml:id="echoid-s14677" xml:space="preserve"> & erit æquidi-
              <lb/>
            diſtans z y cõtingenti [per 28 p 1:</s>
            <s xml:id="echoid-s14678" xml:space="preserve"> quia anguli interiores ad c & z ſunt recti:</s>
            <s xml:id="echoid-s14679" xml:space="preserve"> ille per fabricationem,
              <lb/>
            hic per 18 p 3.</s>
            <s xml:id="echoid-s14680" xml:space="preserve">] Quare z y eſt perpendicularis ſuper
              <lb/>
              <figure xlink:label="fig-0216-01" xlink:href="fig-0216-01a" number="186">
                <variables xml:id="echoid-variables175" xml:space="preserve">a e t o f z h g d j c p k b q r</variables>
              </figure>
            ſuperficiem a c z [per 8 p 11.</s>
            <s xml:id="echoid-s14681" xml:space="preserve">] Quare t q non eſt per-
              <lb/>
            pendicularis ſuper eandem ſuperficiem.</s>
            <s xml:id="echoid-s14682" xml:space="preserve"> Verùm
              <lb/>
            quoniam k eſt p olus circuli b r z:</s>
            <s xml:id="echoid-s14683" xml:space="preserve"> [quia eſt in axe co
              <lb/>
            nico per fabricationem] palàm, cum lineæ k o, k z
              <lb/>
            ſint æquales [per 5 defin.</s>
            <s xml:id="echoid-s14684" xml:space="preserve"> 1 ſphæricorum Theodo-
              <lb/>
            ſij,] & axis a k communis, & a o æqualis a z [per 18
              <lb/>
            d 11:</s>
            <s xml:id="echoid-s14685" xml:space="preserve"> quia utraque eſt latus conicum] quòd erit an-
              <lb/>
            gulus a o k æqualis angulo a z k [per 8 p 1] & ita an-
              <lb/>
            gulus a z k rectus:</s>
            <s xml:id="echoid-s14686" xml:space="preserve"> [quia a o k illi ęqualis, rectus eſt:</s>
            <s xml:id="echoid-s14687" xml:space="preserve">
              <lb/>
            cum k o ſit perpendicularis a o per fabricationem.</s>
            <s xml:id="echoid-s14688" xml:space="preserve">]
              <lb/>
            Cum ergo linea k z ſit perpẽdicularis ſuper a z, quæ
              <lb/>
            eſt linea longitudinis:</s>
            <s xml:id="echoid-s14689" xml:space="preserve"> erit perpendicularis ſuper ſu
              <lb/>
            perficiem, contingentem pyramidem, ſuper hanc li
              <lb/>
            neam longitudinis [ut demonſtratum eſt 54 n 5.</s>
            <s xml:id="echoid-s14690" xml:space="preserve">]
              <lb/>
            Sed t q eſt in ſuperficie contingente:</s>
            <s xml:id="echoid-s14691" xml:space="preserve"> quia eſt cõmu-
              <lb/>
            nis ſectio ſuperficiei contingenti & ſectioni.</s>
            <s xml:id="echoid-s14692" xml:space="preserve"> Igitur
              <lb/>
            k z eſt perpendicularis ſuper t q [per 3 d 11.</s>
            <s xml:id="echoid-s14693" xml:space="preserve">] Duca-
              <lb/>
            tur autem h z in ſuperficie ſectionis perpendicula-
              <lb/>
            ris ſuper lineam t q [per 11 p 1.</s>
            <s xml:id="echoid-s14694" xml:space="preserve">] Cum autẽ linea k z
              <lb/>
            ſit extra ſuperficiem ſectionis:</s>
            <s xml:id="echoid-s14695" xml:space="preserve"> ſecabit lineã h z, nec
              <lb/>
            erit una linea [per 1 p 11.</s>
            <s xml:id="echoid-s14696" xml:space="preserve">] Quare illa ſuperficies k z h
              <lb/>
            ſecat ſuperficiem ſectionis, ſuper lineam h z com-
              <lb/>
            munem:</s>
            <s xml:id="echoid-s14697" xml:space="preserve"> & ſecat lineam t q ſuper punctum z:</s>
            <s xml:id="echoid-s14698" xml:space="preserve"> &
              <lb/>
            ſuperficies h z k ſecat ſuperficiem d z k, ſuper lineam communem k z:</s>
            <s xml:id="echoid-s14699" xml:space="preserve"> uerùm d z eſt in ſuperficie ſe-
              <lb/>
            ctionis, & ſecatur à linea k z in puncto z:</s>
            <s xml:id="echoid-s14700" xml:space="preserve"> & punctum t eſt ſupra ſuperficiem k z h, punctum q infra:</s>
            <s xml:id="echoid-s14701" xml:space="preserve"> &
              <lb/>
            ita ſuperficies k z h ſecabit ſuperficiem d z q ſuper lineam communem:</s>
            <s xml:id="echoid-s14702" xml:space="preserve"> & illa linea communis eſt
              <lb/>
            perpendicularis ſuper lineam t q:</s>
            <s xml:id="echoid-s14703" xml:space="preserve"> quia linea illa eſt in ſuperficie h z k, ſuper quam eſt perpendicula-
              <lb/>
            ris t q [ut oſtenſum eſt.</s>
            <s xml:id="echoid-s14704" xml:space="preserve">] Et quoniam ſuperficies h z k ſecat ſuperficiem d z q:</s>
            <s xml:id="echoid-s14705" xml:space="preserve"> & declinatio ſuperfi-
              <lb/>
            ciei h z k à ſuperficie ſectionis fit ex parte z c:</s>
            <s xml:id="echoid-s14706" xml:space="preserve"> erit linea communis ſectioni illarum ſuperficierũ in-
              <lb/>
            ter lineas q z, d z.</s>
            <s xml:id="echoid-s14707" xml:space="preserve"> Et ita concurret cum perpendiculari ſub axe.</s>
            <s xml:id="echoid-s14708" xml:space="preserve"> Et quòd neceſſariò concurrat, pro-
              <lb/>
            batum eſt in libro quinto [quia anguli e d z, d z p ſunt acuti:</s>
            <s xml:id="echoid-s14709" xml:space="preserve"> ille per theſin, hic, quia pars ẽſt recti
              <lb/>
            t z p.</s>
            <s xml:id="echoid-s14710" xml:space="preserve">] Et ita eſt propoſitum.</s>
            <s xml:id="echoid-s14711" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div502" type="section" level="0" n="0">
          <head xml:id="echoid-head446" xml:space="preserve" style="it">31. Linea recta tota ab uno ſpeculi conici conuexi latere ad uiſum reflecti po-
            <lb/>
          teſt. 41 p 7.</head>
          <p>
            <s xml:id="echoid-s14712" xml:space="preserve">SIt ergo pyramis:</s>
            <s xml:id="echoid-s14713" xml:space="preserve"> cuius uertex a
              <gap/>
            axis a h:</s>
            <s xml:id="echoid-s14714" xml:space="preserve"> linea longitudinis a z.</s>
            <s xml:id="echoid-s14715" xml:space="preserve"> Et à puncto z ducatur perpendi-
              <lb/>
            cularis ſuper ſuperficiem, contingentem pyramidem in linea a z [per 12 p 11] quæ neceſſariò
              <lb/>
            concurret cum axe [per 11 ax.</s>
            <s xml:id="echoid-s14716" xml:space="preserve"> quia angulus h a z eſt acutus per 17 p 1:</s>
            <s xml:id="echoid-s14717" xml:space="preserve"> cum a d z ſit rectus per
              <lb/>
            18 d 11.</s>
            <s xml:id="echoid-s14718" xml:space="preserve">] Sit linea t z h.</s>
            <s xml:id="echoid-s14719" xml:space="preserve"> Ducatur à puncto a linea extra pyramidem, ultra ſuperficiem contingentem
              <lb/>
            pyramidem in linea a z, faciens angulum acutum cum axe & cum linea longitudinis a z:</s>
            <s xml:id="echoid-s14720" xml:space="preserve"> quæ ſit a n.</s>
            <s xml:id="echoid-s14721" xml:space="preserve">
              <lb/>
            Et in ſuperficie a h n à puncto h ducatur linea, cum axe faciens angulum æqualem angulo a h z:</s>
            <s xml:id="echoid-s14722" xml:space="preserve"> quæ
              <lb/>
            linea neceſſariò concurret cum linea a n:</s>
            <s xml:id="echoid-s14723" xml:space="preserve"> [per 11 ax.</s>
            <s xml:id="echoid-s14724" xml:space="preserve"> quia anguli n a h & a h z ex theſi acuti, ſunt mi-
              <lb/>
            nores duobus rectis] quæ ſit h o.</s>
            <s xml:id="echoid-s14725" xml:space="preserve"> Et facto ſuper punctum z circulo æquidiſtante baſi:</s>
            <s xml:id="echoid-s14726" xml:space="preserve"> [ut oſtenſum
              <lb/>
            eſt 52 n 5] tranſibit h o per circulum, ſicut h z tranſit per ipſum.</s>
            <s xml:id="echoid-s14727" xml:space="preserve"> Ducatur linea o z:</s>
            <s xml:id="echoid-s14728" xml:space="preserve"> & producatur
              <lb/>
            ad punctum f.</s>
            <s xml:id="echoid-s14729" xml:space="preserve"> Quoniam linea o z ſecat ſuperficiem, contingentem pyramidem in linea a z:</s>
            <s xml:id="echoid-s14730" xml:space="preserve"> cum li-
              <lb/>
            nea h z ſit perpendicularis ſuper illam ſuperficiem:</s>
            <s xml:id="echoid-s14731" xml:space="preserve"> [per fabricationem] erit angulus o z h maior
              <lb/>
            recto:</s>
            <s xml:id="echoid-s14732" xml:space="preserve"> quia a z h rectus eſt [per fabricationẽ.</s>
            <s xml:id="echoid-s14733" xml:space="preserve">] Igitur [per 13 p 1] angulus f z h acutus.</s>
            <s xml:id="echoid-s14734" xml:space="preserve"> À
              <unsure/>
            puncto z du
              <lb/>
            </s>
          </p>
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