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

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        <div xml:id="echoid-div440" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s12448" xml:space="preserve">
              <pb o="185" file="0191" n="191" rhead="OPTICAE LIBER V."/>
            ſectionisi:</s>
            <s xml:id="echoid-s12449" xml:space="preserve"> & ducatur linea q a.</s>
            <s xml:id="echoid-s12450" xml:space="preserve"> Palàm, quòd angulus b e h æqualis eſt angulo e n a [per 29 p 1:</s>
            <s xml:id="echoid-s12451" xml:space="preserve"> quia
              <lb/>
            a n, h e ſunt parallelæ per fabricationem] & angulus h e a æqualis angulo e a n:</s>
            <s xml:id="echoid-s12452" xml:space="preserve"> & [per 12 n 4] angu
              <lb/>
            lus b e h æqualis angulo h e a:</s>
            <s xml:id="echoid-s12453" xml:space="preserve"> erit angulus e a n æqualis angulo e n a:</s>
            <s xml:id="echoid-s12454" xml:space="preserve"> quare [per 6 p 1] e n æqualis
              <lb/>
            e a, & e q perpendicularis:</s>
            <s xml:id="echoid-s12455" xml:space="preserve"> [duabus rectis e a, e n per 3 d 11:</s>
            <s xml:id="echoid-s12456" xml:space="preserve"> quia perpendicularis eſt per fabricationẽ
              <lb/>
            triangulo a e n] erit [per 4 p 1] triangulũ q e a æquale triangulo q e n:</s>
            <s xml:id="echoid-s12457" xml:space="preserve"> & erit q n æqualis q a:</s>
            <s xml:id="echoid-s12458" xml:space="preserve"> & erit
              <lb/>
            [per 5 p 1] q n a æqualis angulo q a n:</s>
            <s xml:id="echoid-s12459" xml:space="preserve"> ſed angulus t q i æqualis angulo q n a, & angulus i q a æqua-
              <lb/>
            lis angulo q a n:</s>
            <s xml:id="echoid-s12460" xml:space="preserve"> [per 29 p 1:</s>
            <s xml:id="echoid-s12461" xml:space="preserve"> quia q i, a n ſunt parallelæ per fabricationem] erit angulus i q t æqualis
              <lb/>
            angulo i q a.</s>
            <s xml:id="echoid-s12462" xml:space="preserve"> Quare a reflectetur adt à puncto columnæ, quod eſt q [per 12 n 4.</s>
            <s xml:id="echoid-s12463" xml:space="preserve">] Eodem modo pro-
              <lb/>
            babitur, quòd reflectetur a ad t â punctis l, m:</s>
            <s xml:id="echoid-s12464" xml:space="preserve"> Et ita à tribus punctis columnæ ex eadem parte:</s>
            <s xml:id="echoid-s12465" xml:space="preserve"> nec
              <lb/>
            poteſt à pluribus.</s>
            <s xml:id="echoid-s12466" xml:space="preserve"> Detur enim aliud:</s>
            <s xml:id="echoid-s12467" xml:space="preserve"> ducto latere [cylindri, ut oſtenſum eſt 47 n] ab illo puncto:</s>
            <s xml:id="echoid-s12468" xml:space="preserve">
              <lb/>
            cadet in circulum, quem habemus:</s>
            <s xml:id="echoid-s12469" xml:space="preserve"> & probabitur, quòd à pũcto caſus, qui eſt in circulo, poterit re-
              <lb/>
            flecti a ad t, repetita ꝓbatione:</s>
            <s xml:id="echoid-s12470" xml:space="preserve"> quod eſt impoſsibile [ut oſtẽſum eſt 86 n.</s>
            <s xml:id="echoid-s12471" xml:space="preserve">] Ex arcu uerò circuli op-
              <lb/>
            poſito [arcui g d e] poterit reflecti a ad b ab uno puncto:</s>
            <s xml:id="echoid-s12472" xml:space="preserve"> [per 73 n] ſit illud z:</s>
            <s xml:id="echoid-s12473" xml:space="preserve"> & ducatur diameter
              <lb/>
            h z:</s>
            <s xml:id="echoid-s12474" xml:space="preserve"> & [per 31 p 1] ei æquidiſtans a s:</s>
            <s xml:id="echoid-s12475" xml:space="preserve"> & ducatur b z:</s>
            <s xml:id="echoid-s12476" xml:space="preserve"> quæ concurrat cum a s in puncto s:</s>
            <s xml:id="echoid-s12477" xml:space="preserve"> [cõcurret au-
              <lb/>
            tem per lemma Procli ad 29 p 1] & erigatur perpendicularis:</s>
            <s xml:id="echoid-s12478" xml:space="preserve"> [ſuper circulũ, cuius centrũ eſt h] quę
              <lb/>
            ſit o z:</s>
            <s xml:id="echoid-s12479" xml:space="preserve"> quæ erit latus [per 21 d 11] & [per 6 p 11] æquidiſtans t b:</s>
            <s xml:id="echoid-s12480" xml:space="preserve"> & ducatur t s:</s>
            <s xml:id="echoid-s12481" xml:space="preserve"> quę ſecabitur à linea o
              <lb/>
            z:</s>
            <s xml:id="echoid-s12482" xml:space="preserve"> [per lemma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s12483" xml:space="preserve">] Sit ſectio in pũcto o.</s>
            <s xml:id="echoid-s12484" xml:space="preserve"> Probabitur modo prędicto, quòd a reflectetur
              <lb/>
            ad t à puncto o.</s>
            <s xml:id="echoid-s12485" xml:space="preserve"> Et ſi ſumatur exilla parte punctum aliud columnæ, à quo poſsit reflecti:</s>
            <s xml:id="echoid-s12486" xml:space="preserve"> per repli-
              <lb/>
            cationem probationis probabitur, quòd ab alio puncto circuli, quàm z, poteſt reflecti ex parte illa:</s>
            <s xml:id="echoid-s12487" xml:space="preserve">
              <lb/>
            quod eſt impoſsibile [ut demonſtratũ eſt 75 n.</s>
            <s xml:id="echoid-s12488" xml:space="preserve">] Si ergo a ab uno puncto circuli reflectitur ad b ex
              <lb/>
            aliqua parte:</s>
            <s xml:id="echoid-s12489" xml:space="preserve"> reflectetur ab uno columnæ ex eadem ad t:</s>
            <s xml:id="echoid-s12490" xml:space="preserve"> ſi à duobus, à duobus:</s>
            <s xml:id="echoid-s12491" xml:space="preserve"> ſi à tribus, à tribus:</s>
            <s xml:id="echoid-s12492" xml:space="preserve">
              <lb/>
            nec poteſt amplius ab illa parte:</s>
            <s xml:id="echoid-s12493" xml:space="preserve"> ab oppoſita uerò parte non niſi ab uno puncto circuli tantùm, &
              <lb/>
            ab uno columnæ tantùm.</s>
            <s xml:id="echoid-s12494" xml:space="preserve"> Item t b æquidiſtat u h:</s>
            <s xml:id="echoid-s12495" xml:space="preserve"> [ut ab initio demõſtratum eſt:</s>
            <s xml:id="echoid-s12496" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s12497" xml:space="preserve"> per 35 d 1 ſunt
              <lb/>
            in eadem ſuperficie, quæ eſt t b u h] nec poteſt ſumi ſuperficies æqualis, in qua ſit t cum u h, præter
              <lb/>
            ſuperficiem t b u h.</s>
            <s xml:id="echoid-s12498" xml:space="preserve"> Similiter non poteſt ſuperficies ſumi, in qua ſit a cum u h, præter ſuperficiem a u
              <lb/>
            h, quæ eſt perpendicularis [circulo, cuius cẽtrum h, per 18 p 11.</s>
            <s xml:id="echoid-s12499" xml:space="preserve">] Igitur t non eſt in eadem ſuperficie
              <lb/>
            perpendiculari cum a, necin eodem circulo, nec eſt in axe, quia eſt in linea ei æ quidiſtante.</s>
            <s xml:id="echoid-s12500" xml:space="preserve"> Super-
              <lb/>
            ficies igitur, in qua a reflectitur ad t, eft ſectio colũnaris [per 9 th.</s>
            <s xml:id="echoid-s12501" xml:space="preserve"> Sereni de ſectione cylindri.</s>
            <s xml:id="echoid-s12502" xml:space="preserve">] Ve-
              <lb/>
            rùm producta ſit t a ultra t, & a ex utraq;</s>
            <s xml:id="echoid-s12503" xml:space="preserve"> parte:</s>
            <s xml:id="echoid-s12504" xml:space="preserve"> & ſit r p.</s>
            <s xml:id="echoid-s12505" xml:space="preserve"> Cum quatuor ſint ſuperficies reflexionis:</s>
            <s xml:id="echoid-s12506" xml:space="preserve">
              <lb/>
            quia à quatuor punctis [q, l, m;</s>
            <s xml:id="echoid-s12507" xml:space="preserve"> o] ſit reflexio, & in qualibet harum ſint duo puncta t, a:</s>
            <s xml:id="echoid-s12508" xml:space="preserve"> erit r p com-
              <lb/>
            munis quatuor ſuperficiebus reflexionis:</s>
            <s xml:id="echoid-s12509" xml:space="preserve"> [per 1 p 11:</s>
            <s xml:id="echoid-s12510" xml:space="preserve"> quia uiſus & uiſibile, quæ ſunt in linea r p, ſunt
              <lb/>
            in qualibet reflexionis ſuperficie per 23 n 4] & quælibet harũ ſuperficierum ſecat ſuperficiem, con-
              <lb/>
            tingentem ſpeculum in puncto ſ
              <gap/>
            æ reflexionis, ſuper ſuam lineam communem, nõ ſuper eandem
              <lb/>
            [quia cum puncta reflexionis ſint diuerſa, etiam communes ſectiones illarum ſuperficierum (quæ
              <lb/>
            ſuntrectæ lineæ per 3 d 11) diuerſæ erunt.</s>
            <s xml:id="echoid-s12511" xml:space="preserve">] Linea ergo r p perpendicularis eſt ſuper unam linearum
              <lb/>
            quatuor cõmunium, non ſuper duas:</s>
            <s xml:id="echoid-s12512" xml:space="preserve"> eſſet enim perpendicularis ſuper ſuperficiem contingentem:</s>
            <s xml:id="echoid-s12513" xml:space="preserve">
              <lb/>
            [per 3 d 11] & ita perueniret ad axem.</s>
            <s xml:id="echoid-s12514" xml:space="preserve"> [Quia enim per 21 d 11 latus cylindraceum ęquidiſtat axi:</s>
            <s xml:id="echoid-s12515" xml:space="preserve"> & r p
              <lb/>
            perpendicularis plano tangenti ex cõcluſo, ſimul perpendicularis eſt lateri per 3 d 11:</s>
            <s xml:id="echoid-s12516" xml:space="preserve"> ergo per lem-
              <lb/>
            ma Procli ad 29 p 1 r p (quæ paulo antè oſtẽſa eſt extra axẽ eſſe) cõtinuata ſecabit axẽ;</s>
            <s xml:id="echoid-s12517" xml:space="preserve"> quod eſt ab:</s>
            <s xml:id="echoid-s12518" xml:space="preserve">
              <lb/>
            ſurdum.</s>
            <s xml:id="echoid-s12519" xml:space="preserve">] Sunt ergo diuerſæ perpendiculares à puncto t ad has quatuor lineas communes:</s>
            <s xml:id="echoid-s12520" xml:space="preserve"> nec eſt
              <lb/>
            niſi una perpendicularis tantùm, quę tranſit per a.</s>
            <s xml:id="echoid-s12521" xml:space="preserve"> Et perpendicularis aut eſt æquidiſtans lineæ re-
              <lb/>
            flexionis:</s>
            <s xml:id="echoid-s12522" xml:space="preserve"> aut concurrit cum ea ultra ſpeculum, uel intra.</s>
            <s xml:id="echoid-s12523" xml:space="preserve"> Si fuerit æquidiſtans:</s>
            <s xml:id="echoid-s12524" xml:space="preserve"> erit locus imaginis
              <lb/>
            punctum reflexionis, ut probatum eſt
              <gap/>
            [91.</s>
            <s xml:id="echoid-s12525" xml:space="preserve"> n] Et cum quatuor ſint reflexionis puncta:</s>
            <s xml:id="echoid-s12526" xml:space="preserve"> erunt qua-
              <lb/>
            tuor imagines.</s>
            <s xml:id="echoid-s12527" xml:space="preserve"> Si concurrit, cum quatuor ſunt perpendiculares:</s>
            <s xml:id="echoid-s12528" xml:space="preserve"> erunt concurſus quatuor, & qua-
              <lb/>
            tuor im agines.</s>
            <s xml:id="echoid-s12529" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div442" type="section" level="0" n="0">
          <head xml:id="echoid-head399" xml:space="preserve" style="it">96. Viſu & uiſibili datis, in ſpeculo cylindraceo cauo punctum reflexionis inuenire. 16 p 9.</head>
          <p>
            <s xml:id="echoid-s12530" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s12531" xml:space="preserve"> datis puncto uiſo, & puncto uiſus:</s>
            <s xml:id="echoid-s12532" xml:space="preserve"> erit inuenire punctum reflexionis.</s>
            <s xml:id="echoid-s12533" xml:space="preserve"> Verbi gratia;</s>
            <s xml:id="echoid-s12534" xml:space="preserve">
              <lb/>
            ſit a punctum uiſum:</s>
            <s xml:id="echoid-s12535" xml:space="preserve"> b centrum uiſus.</s>
            <s xml:id="echoid-s12536" xml:space="preserve"> Fiat ſuperficies ſecans columnam æquidiſtanter baſi
              <lb/>
            [ut oſtenſum eſt 47 n] trànſiens per a:</s>
            <s xml:id="echoid-s12537" xml:space="preserve"> & [per 5 th.</s>
            <s xml:id="echoid-s12538" xml:space="preserve"> Sereni de ſectione cylindri] faciet circu-
              <lb/>
            lum.</s>
            <s xml:id="echoid-s12539" xml:space="preserve"> b aut eſt in ſuperficie huius circuli:</s>
            <s xml:id="echoid-s12540" xml:space="preserve"> aut nõ.</s>
            <s xml:id="echoid-s12541" xml:space="preserve"> Si fuerit:</s>
            <s xml:id="echoid-s12542" xml:space="preserve"> inueniemus punctũ reflexionis in illo cir-
              <lb/>
            culo, ſicut dictum eſt in ſphærico concauo [73 n.</s>
            <s xml:id="echoid-s12543" xml:space="preserve">] Si nõ fuerit:</s>
            <s xml:id="echoid-s12544" xml:space="preserve"> ducatur [per 11 p 11] à puncto b perpẽ-
              <lb/>
            dicularis ſuper ſuperficiem huius circuli:</s>
            <s xml:id="echoid-s12545" xml:space="preserve"> & replicetur ſuprà dicta probatio:</s>
            <s xml:id="echoid-s12546" xml:space="preserve"> & inuenietur pũctum
              <lb/>
            reflexionis.</s>
            <s xml:id="echoid-s12547" xml:space="preserve"> Duplici autem uiſu adhibito, una imago in ueritate, efficientur duæ, ſed contiguæ uel
              <lb/>
            admixtæ:</s>
            <s xml:id="echoid-s12548" xml:space="preserve"> unde uidebitur una.</s>
            <s xml:id="echoid-s12549" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div443" type="section" level="0" n="0">
          <head xml:id="echoid-head400" xml:space="preserve" style="it">97. Cõmunis ſectio ſuperficierũ, reflexionis & ſpeculiconici caui eſt latus coni, aut ellipſis. 2 p 9.</head>
          <p>
            <s xml:id="echoid-s12550" xml:space="preserve">IN ſpeculis pyramidalibus concauis linea, communis ſuperficiei reflexionis & ſuperficiei ſpecu
              <lb/>
            li, aut erit linea lõgitudinis ſpeculi:</s>
            <s xml:id="echoid-s12551" xml:space="preserve"> aut erit ſectio pyramidalis.</s>
            <s xml:id="echoid-s12552" xml:space="preserve"> Si fuerit linea longitudinis:</s>
            <s xml:id="echoid-s12553" xml:space="preserve"> erũt
              <lb/>
            loca imaginum in ipſo ſpeculo.</s>
            <s xml:id="echoid-s12554" xml:space="preserve"> Si fuerit ſectio pyramidalis:</s>
            <s xml:id="echoid-s12555" xml:space="preserve"> erunt loca imaginum aliquando ci-
              <lb/>
            tra uiſum:</s>
            <s xml:id="echoid-s12556" xml:space="preserve"> aliquando in uiſu:</s>
            <s xml:id="echoid-s12557" xml:space="preserve"> aliquando inter uiſum & ſpeculum:</s>
            <s xml:id="echoid-s12558" xml:space="preserve"> & aliquando ultra ſpeculum, ſi-
              <lb/>
            cut oſtenſum eſt in ſpeculo columnari concauo.</s>
            <s xml:id="echoid-s12559" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div444" type="section" level="0" n="0">
          <head xml:id="echoid-head401" xml:space="preserve" style="it">98. Siuiſus ſit in communi ſectione axis & rectæ lineæ perpendicularis plano, ſpeculum co-
            <lb/>
          nicum cauum tangẽti: reflectetur à tota peripheria circuli (cuius centrum eſt dict a communis
            <lb/>
          ſectio) per lineas perpendiculares: & imago uidebitur in centro uiſus. 17 p 9.</head>
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