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 contents

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[211.] 11. Radi{us} ſpeculo perpendicularis, reflectitur in ſeipſum. 11.12 p 5.
[212.] 12. In ſpeculis, conuexis, cauis: ſphærico, conico cylindraceo, anguli incidentiæ & reflexio-nis æquantur. 12.13.14.15.16.17.20 p 5.
[213.] 13. Superficies reflexionis eſt perpendicularis plano ſpeculum in reflexionis puncto tan-genti. 25 p 5.
[214.] 14. Inter uiſibile & ſpeculũ innumer abiles pyramides fiũt alternis baſib. & uerticib{us}. 22 p 5.
[215.] 15. Lux à ſuperficie polita longinquiore reflexa, trifariam debilitatur.
[216.] 16. Lux & color reflectuntur per line{as} phyſic{as}, latitudine quadam prędit{as}. 3 p 2.6 p 5.
[217.] 17. Reflexio lucis & coloris à ſuperficie aſper a facta, plerun fugit uiſum. 1 p 5.
[218.] 18. Radij incidentiæ & reflexionis, ſit{us} ſimilitudine conueniunt. Ita anguli incidentiæ & reflexionis æquantur. 20 p 5.
[219.] 19. Colorem luci permiſtum reflecti, reflexionis organo ostenditur. 3 p 5.
[220.] QVOÒD COMPREHENSIO FORMARVM È CORPORIBVS politis fiat reflexione. Cap. 1111. 20. Falſa eſt utra opinio: & radios à uiſu ad ſpeculum miſſos, inde́ ad uiſibile reflexos, ima ginem percipere: & imaginẽ in ſpeculo iam antè impreſſam inde ad uiſum manare. 23. 24 p 5.
[221.] DE MODO COMPREHENSIONIS FORMARVM È COR-poribus politis. Cap. V. 21. Imago uiſibilis percipitur è reflexione formæ uiſibilis à ſpeculo ad uiſum facta. 24 p 5.
[222.] 22. Si uiſibile & ſpeculum figuræ ſit{us}́ ſimilitudine conueniant: uera & distincta imago uidetur. 35 p 5.
[223.] 23. Superficies reflexionis quatuor habet puncta: uiſibilis: reflexionis: uiſ{us}: & terminũ per-pendicularis ductæ à puncto reflexionis ſuper planum in eodem puncto ſpeculum tangens. Ita perpendicularis hæc cõmunis eſt omnib{us} reflexionis ſuperficieb{us}. 27 p 5.6 p 6.24 p 7.3 p 8.3 p 9.
[224.] 24. Si uiſ{us} ſit extra ſuperficiem ſpeculi ſphærici conuexi, uelipſi continuam: communis ſe-ctio baſis pyramidis opticæ & ſuperficiei ſpeculi, erit peripheria minimi in ſphæra circuli. 3 p 6.
[225.] 25. Si duarum rectarum linearum à uiſu, alter a ſpeculum ſphæricum conuexum tangat, re-liqua per centrum ſecet: tangens circa ſecantem fixam cõuerſa, definiet ſegmentum ſuperficiei ſpeculι: à cui{us} puncto quolibet poteſt ad uiſum fieri reflexio. Et centra uiſ{us} & ſpeculi, puncta reflexionis & uiſibilis ſunt in reflexionis ſuperficie. 2.5.6 p 6.
[226.] 26. Siduo plana à cẽtro uiſiis, ducãtur ք later a cõſpicuam ſpeculi cylindracei cõuexi ſuperficiẽ terminãtia: tangẽt ſpeculũ: & facient in uiſu cõmunem ſectionẽ par allelã axiſpeculi. 2.3 p 7.
[227.] 27. Si linea recta à cẽtro uiſ{us}, ducta ad punctũ cõſpicuæ ſuper-ficiei ſpeculi cylindr acei cõuexi, cõtinuetur: ſecabit ſpeculũ. 4.5 p 7.
[228.] 28. In ſpeculo cylindraceo conuexo, à quolibet conſpicuæ ſuperficiei puncto poteſt ad uiſum reflexio fieri. 25 p 7.
[229.] 29. Si uiſ{us} ſit extra ſuperficiem ſpeculi cylindr acei conuexi, in plano uiſibilis per axem du-cto: cõm unis ſectio ſuperficier um reflexionis & ſpeculi, erit lat{us} cylindri: & unicum tantùm eſt in eadem conſpicua ſuperficie planum, à quo ad eundem uiſum reflexio fieri poteſt. 7.16 p 7.
[230.] 30. Si uiſ{us} ſit extrá ſuperficiem ſpeculi cylindracei cõuexi, in planò uiſibilis ad axem recto: communis ſectio ſuperficierum reflexionis & ſpeculi, erit circul{us}: & unic{us} tantùm eſt in ea-dem conſpicuà ſuperficie, à quo ad uiſum reflexio fieri poteſt. 9.17 p 7.
[231.] 31. Si uiſ{us} ſit extra ſuperficiem ſpeculi cylindracei conuexi, in plano uiſibilis ad axem obli-quo: communis ſectio ſuperficierum reflexionis & ſpeculi erit ellipſis: & plures in eadem conſpi-cua ſuperficie eſſe poſſunt, à quib{us} ad eundem uiſum reflexio fiat. 10. 18 p 7.
[232.] 32. Si communis ſectio ſuperficierum reflexionis & ſpeculi cylindr acei conuexi, fuerit lat{us} cylindri, uel cιrcul{us}: reflexio à quocun communis ſectionis puncto facta, in eadem ſuperficie ſemper fiet. 19. 20 p 7.
[233.] 33. Ab uno cõmunis ſectionis ſuperficierum reflexionis & ſpeculi cylindr acei conuexi pun-cto, unum uiſibilis punctum ad unum uiſum in eadem ſuperficie reflectitur. 22 p 7.
[234.] 34. Si rect a line à reflexionis puncto, ſit perpendicularis ſpeculo cylindraceo conuexo: in-t{us} continuata, tranſibit per centrum circuli baſib{us} par alleli: & contrà. 21 p 7.
[235.] 35. Si à uiſu extra ſpeculi conici conuexirecti ſuperficiem, uel ipſi continuam ſito, recta li-nea cum uertice axis acutum angulũ faciat: duo plana educta per rect{as} à uiſu, ſpeculum tan-gentes & conica latera, per tact{us} puncta tranſeuntia, tangent ſpeculum, & cõſpicuam ſuper-ficiem dimidiat a minorem, à qua ad uiſum reflexio fiat, terminabunt. 1. 2 p 7.
[236.] 36. Si à uiſu recta linea, ſit perpendicularis uertici axis ſpecu- li conici cõuexi recti: duo plana educta per rect{as} ſpeculum in ter- minis diametricirculi, ad baſim paralleli tangentes, & later a co- nica per tact{us} puncta tranſeuntia: tangent ſpeculum: & dimi- diatam ſuperficiem conſpicuam, à qua ad uiſum reflexio fiat, ter- minabunt. 89 p 4.
[237.] 37. Si recta linea à centro uiſ{us}, cum uertice ſpeculi conici conuexi recti angulum obtuſum faciens, continuata concurr at extra ſpeculum, cum diametro circuli ad baſim par alleli conti-nuata: duo plana educta per rect{as} à concurſu ſpeculum in dicto circulo tangentes, & later a conica per tact{us} puncta tranſeuntia, tangent ſpeculum: & ſuperficiem conſpicuam dimidiata maiorem, à qua ad uiſum reflexio fiat: terminabunt. 90 p 4.
[238.] 38. Sirecta linea à uiſu per uerticem ſpeculi conici conuexi recti, continuetur cum conico latere: tota ſuperficies, præter dictum lat{us}, uidebitur. 91 p 4.
[239.] 39. Si recta linea à uiſu in uerticem ſpeculi conici conuexi recti, continuetur cum axe: tota ſuperficies conica uidebitur. 92 p 4.
[240.] 40. Si communis ſectio ſuperficierum, reflexionis & ſpeculi conici conuexi fuerit lat{us} coni-cum: à quolιbet conſpicuæ ſuperficiei puncto ad uiſum reflexio fieri poteſt. 31 p 7.
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            <s xml:id="echoid-s14932" xml:space="preserve">
              <pb o="213" file="0219" n="219" rhead="OPTICAE LIBER VI."/>
            ſus [per 13 p 1.</s>
            <s xml:id="echoid-s14933" xml:space="preserve">] Igitur angulus e n d eſt acutus [per 32 p 1.</s>
            <s xml:id="echoid-s14934" xml:space="preserve">] Et ſit linea c x cõtingens ſectionẽ in pun
              <lb/>
            cto c.</s>
            <s xml:id="echoid-s14935" xml:space="preserve"> Patet ergo, ut in prædicta figura [30 n] quòd angulus d c x eſt obtuſus:</s>
            <s xml:id="echoid-s14936" xml:space="preserve"> & qđ perpẽdicularis
              <lb/>
            extracta ex c ſuper c x, ſecabit angulũ d c x:</s>
            <s xml:id="echoid-s14937" xml:space="preserve"> & cõcurret cũ e d ſub d.</s>
            <s xml:id="echoid-s14938" xml:space="preserve"> Ergo hæc perpendicularis ſecet
              <lb/>
            e d in s.</s>
            <s xml:id="echoid-s14939" xml:space="preserve"> Perpẽdicularis ergo extracta ex n ſuք lineã cõtingentẽ ſectionẽ, ſecabit ſectionẽ ultra s:</s>
            <s xml:id="echoid-s14940" xml:space="preserve"> ſed
              <lb/>
            remotius à d quã s:</s>
            <s xml:id="echoid-s14941" xml:space="preserve"> nã iſtę perpendiculares cõcurrent ultra circũferentiã ſectionis.</s>
            <s xml:id="echoid-s14942" xml:space="preserve"> Perpẽdicularis
              <lb/>
            ergo extracta ex puncto n ſuper lineã contingentẽ ſectionẽ, non ſecabit angulũ d c x:</s>
            <s xml:id="echoid-s14943" xml:space="preserve"> erit ergo
              <gap/>
            e-
              <lb/>
            motior ab n e, quàm ſit n d.</s>
            <s xml:id="echoid-s14944" xml:space="preserve"> Ergo hæc perpendicularis ſecat a d ſupra d.</s>
            <s xml:id="echoid-s14945" xml:space="preserve"> Sit ergo perpẽdicularis ex-
              <lb/>
            tracta ex n ſuper lineam cõtingentẽ ſectionẽ, linea n q.</s>
            <s xml:id="echoid-s14946" xml:space="preserve"> Et r e ſecat e n, & ſecat circumferẽtiã ſectio-
              <lb/>
            nis:</s>
            <s xml:id="echoid-s14947" xml:space="preserve"> & eſt in ſuperficie eius:</s>
            <s xml:id="echoid-s14948" xml:space="preserve"> & n q eſt in ſuperficie ſectionis.</s>
            <s xml:id="echoid-s14949" xml:space="preserve"> Si ergo r e extrahatur rectè, ſecabit n q
              <lb/>
            [quia cõtinuata ſecat angulũ n e q.</s>
            <s xml:id="echoid-s14950" xml:space="preserve">] Secet ergo in y:</s>
            <s xml:id="echoid-s14951" xml:space="preserve"> & ſuperficies a n d ſecabit ſuperficiẽ ſectionis.</s>
            <s xml:id="echoid-s14952" xml:space="preserve">
              <lb/>
            Itẽ quia punctũ e eſt extra ſuperficiẽ a n d:</s>
            <s xml:id="echoid-s14953" xml:space="preserve"> (nã ſuperficies a n d nõ eſt ſuperficies ſectionis [in qua
              <lb/>
            eſt punctũ e] quia punctũ a eſt extra ſuperficiẽ ſectionis:</s>
            <s xml:id="echoid-s14954" xml:space="preserve"> & quia a e eſt perpẽdicularis ſuper ſuperfi
              <lb/>
            ciẽ ſectionis, & e eſt in circumferentia illius) ergo n c d eſt differentia cõmunis ſuperficiei a n d &
              <lb/>
            ſuperficiei ſectionis:</s>
            <s xml:id="echoid-s14955" xml:space="preserve"> & n q concurrit cũ ſectione ultra c [ut patuit.</s>
            <s xml:id="echoid-s14956" xml:space="preserve">] Ergo n q eſt ultra ſuperficiem
              <lb/>
            a n d:</s>
            <s xml:id="echoid-s14957" xml:space="preserve"> y ergo eſt ultra lineam a p g [quæ nõ eſt in ſuperficie a n d.</s>
            <s xml:id="echoid-s14958" xml:space="preserve">] Si ergo uiſus fuerit in r, & forma
              <lb/>
            alicuius uiſibilis reflectatur à linea longitudinis:</s>
            <s xml:id="echoid-s14959" xml:space="preserve"> tunc p erit imago o:</s>
            <s xml:id="echoid-s14960" xml:space="preserve"> [per 4 n 5] & y erit imago n:</s>
            <s xml:id="echoid-s14961" xml:space="preserve">
              <lb/>
            & a uidebitur in ſuo loco:</s>
            <s xml:id="echoid-s14962" xml:space="preserve"> quia eſt in uertice pyramidis.</s>
            <s xml:id="echoid-s14963" xml:space="preserve"> Et erit imago lineæ a o n linea tranſiens per
              <lb/>
            pũcta a, p, y:</s>
            <s xml:id="echoid-s14964" xml:space="preserve"> ſed hæc linea eſt cõuexa:</s>
            <s xml:id="echoid-s14965" xml:space="preserve"> quia eſt ultra lineã a p g.</s>
            <s xml:id="echoid-s14966" xml:space="preserve"> Sit ergo linea a p y.</s>
            <s xml:id="echoid-s14967" xml:space="preserve"> Et patuitiã, quòd
              <lb/>
            formæ omniũ punctorũ, quæ ſunt in a n, reflectantur ad r ex a e.</s>
            <s xml:id="echoid-s14968" xml:space="preserve"> Lineæ ergo radiales, per quas refle
              <lb/>
            ctuntur illæ formæ, ſunt in ſuperficie trianguli r a e.</s>
            <s xml:id="echoid-s14969" xml:space="preserve"> Omnes ergo imagines lineæ a n ſunt in hac ſu-
              <lb/>
            perficie.</s>
            <s xml:id="echoid-s14970" xml:space="preserve"> Ergo linea a p y conuexa eſt in hac ſuperficie:</s>
            <s xml:id="echoid-s14971" xml:space="preserve"> & p eſt propinquius r quàm y.</s>
            <s xml:id="echoid-s14972" xml:space="preserve"> Et erit
              <lb/>
            conuexitas imaginis huius ex parte uiſus:</s>
            <s xml:id="echoid-s14973" xml:space="preserve"> & erit conuexitas parua:</s>
            <s xml:id="echoid-s14974" xml:space="preserve"> & diameter huius imaginis e-
              <lb/>
            rit minor ipſa linea, modica quantitate.</s>
            <s xml:id="echoid-s14975" xml:space="preserve"> Imagines ergo linearum rectarum, quæ extrahuntur ex
              <lb/>
            uertice pyramidis obliquè ſuper axem:</s>
            <s xml:id="echoid-s14976" xml:space="preserve"> comprehenduntur à uiſu in tali ſpeculo conuexæ.</s>
            <s xml:id="echoid-s14977" xml:space="preserve"> Et for-
              <lb/>
            mę harum linearum reflectuntur à lineis rectis extenſis in longitudine pyramidis.</s>
            <s xml:id="echoid-s14978" xml:space="preserve"> Et hoc eſt, quod
              <lb/>
            uoluimus declarare.</s>
            <s xml:id="echoid-s14979" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div506" type="section" level="0" n="0">
          <head xml:id="echoid-head448" xml:space="preserve" style="it">33. Si recta linea ſit parallela latitudini ſpeculi conici conuexi: & uiſ{us} ſit extra planum di-
            <lb/>
          ctæ lineæ baſi parallelum: reflectetur ab ellipſi: & imago uidebitur maximè curua. 56 p 7.</head>
          <p>
            <s xml:id="echoid-s14980" xml:space="preserve">FOrmæ uerò linearũ æquidiſtantiũ latitudini ſpeculi pyramidalis cõuexi, reflectuntur à lineis
              <lb/>
            conuexis in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s14981" xml:space="preserve"> & conuexitas harum linearum patet, ut in ſpeculo columnari
              <lb/>
            conuexo [29 n.</s>
            <s xml:id="echoid-s14982" xml:space="preserve">] Et per illam eandem uiam etiam ſimiliter patebit, quòd imagines harum li-
              <lb/>
            nearum erunt nimium cõuexæ & manifeſtæ ſenſui.</s>
            <s xml:id="echoid-s14983" xml:space="preserve"> Et erit centrum uiſus extra ſuperficies, in qui-
              <lb/>
            bus eſt cõuexitas formarum harum linearum.</s>
            <s xml:id="echoid-s14984" xml:space="preserve"> Et erunt diametri imaginum harum linearum mul-
              <lb/>
            tò minores ipſis lineis.</s>
            <s xml:id="echoid-s14985" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div507" type="section" level="0" n="0">
          <head xml:id="echoid-head449" xml:space="preserve" style="it">34. Si recta linea nec uertici ſpeculi conici conuexi obliquè incidat, nec latitudini ei{us} ſit paral
            <lb/>
          lela: imaginem uariæ obliquitatis prouario ſit u uiſui offeret. 57 p 7.</head>
          <p>
            <s xml:id="echoid-s14986" xml:space="preserve">DE lineis uerò obliquis exiſtentibus inter hos duos modos, quę appropinquant in ſuo motu
              <lb/>
            lineis extenſis in longitudine pyramidis, habent formas parũ conuexas:</s>
            <s xml:id="echoid-s14987" xml:space="preserve"> quę uerò appropin
              <lb/>
            quant lineis æquidiſtantibus latitudini pyramidis, habent formas manifeſtè conuexas.</s>
            <s xml:id="echoid-s14988" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div508" type="section" level="0" n="0">
          <head xml:id="echoid-head450" xml:space="preserve" style="it">35. In ſpeculo conico conuexo imago conica uidetur. 58 p 7. 40 p 6.</head>
          <p>
            <s xml:id="echoid-s14989" xml:space="preserve">SEd tamen lineæ tortuoſæ, quæ appropinquant uertici pyramidis, habent formas minores, &
              <lb/>
            ſtrictiores & conuexiores.</s>
            <s xml:id="echoid-s14990" xml:space="preserve"> Quæ uerò appropinquant baſi pyramidis, habent formas amplio-
              <lb/>
            res, propter illud, quod declaratum fuit in ſpeculus ſphæricis conuexis:</s>
            <s xml:id="echoid-s14991" xml:space="preserve"> ſcilicet quòd quantò
              <lb/>
            minus fuerit ſpeculum, tantò minores erunt circuli, qui cadunt in ſuperficiem eius:</s>
            <s xml:id="echoid-s14992" xml:space="preserve"> & ſic ima-
              <lb/>
            gines erunt propinquiores centro:</s>
            <s xml:id="echoid-s14993" xml:space="preserve"> idcirco erunt minores.</s>
            <s xml:id="echoid-s14994" xml:space="preserve"> Et ſimiliter ſectiones, quæ cadunt
              <lb/>
            in ſpeculũ pyramidale, quæ ſunt ex parte uerticis pyramidis, ſunt ſtrictiores & minores:</s>
            <s xml:id="echoid-s14995" xml:space="preserve"> & ſic ima-
              <lb/>
            go erit propinquior puncto, in quo cõcurrunt perpendiculares, exeuntes à linea uiſibili perpendi-
              <lb/>
            culariter ſuper lineas contingentes ſectiones, quæ ſunt differentiæ communes:</s>
            <s xml:id="echoid-s14996" xml:space="preserve"> & ideo iſtę ima-
              <lb/>
            gines erunt minores.</s>
            <s xml:id="echoid-s14997" xml:space="preserve"> Sectiones uerò, quæ ſunt ex parte baſis pyramidis, è contrario.</s>
            <s xml:id="echoid-s14998" xml:space="preserve"> Vnde ac-
              <lb/>
            cidit, ut forma comprehenſa in ſpeculo pyramidali conuexo ſit pyramidata:</s>
            <s xml:id="echoid-s14999" xml:space="preserve"> quod ſcilicet fuerit ex
              <lb/>
            parte uerticis ſpeculi, erit ſtrictius, & quod ex parte baſis, erit amplius:</s>
            <s xml:id="echoid-s15000" xml:space="preserve"> & conuexitas latitudinis
              <lb/>
            formæ erit manifeſta.</s>
            <s xml:id="echoid-s15001" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div509" type="section" level="0" n="0">
          <head xml:id="echoid-head451" xml:space="preserve" style="it">36. Imago uiſibilis propinqui ſpeculo conico conuexo, maior: longinqui, minor uidetur. 59 p 7.</head>
          <p>
            <s xml:id="echoid-s15002" xml:space="preserve">ET accidit etiam in his ſpeculis, quòd quantò magis res uiſa appropinquauerit ſpeculo, tantò
              <lb/>
            uidebitur maior:</s>
            <s xml:id="echoid-s15003" xml:space="preserve"> & quantò magis erit remota, tantò uidebitur minor.</s>
            <s xml:id="echoid-s15004" xml:space="preserve"> Fallaciæ ergo, quæ ac-
              <lb/>
            cidunt in his ſpeculis, ſunt ſimiles in omnibus diſpoſitionibus, illis, quæ accidunt in ſpecu-
              <lb/>
            lis columnaribus conuexis, præterquam in pyramidatione formæ.</s>
            <s xml:id="echoid-s15005" xml:space="preserve"/>
          </p>
        </div>
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