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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          <p>
            <s xml:id="echoid-s14734" xml:space="preserve">
              <pb o="211" file="0217" n="217" rhead="OPTICAE LIBER VI."/>
            catur contingens circulum [per 17 p 3] quæ ſit m z:</s>
            <s xml:id="echoid-s14735" xml:space="preserve"> & à puncto f ducatur perpẽdicularis ſupera z,
              <lb/>
            [per 12 p 1] cadens in punctũ eius e:</s>
            <s xml:id="echoid-s14736" xml:space="preserve"> quæ producta cõcurrat cũ a o [per 11 ax.</s>
            <s xml:id="echoid-s14737" xml:space="preserve">] quoniã angulus o a z
              <lb/>
            eſt acutus [ex theſi.</s>
            <s xml:id="echoid-s14738" xml:space="preserve">] Concurrat
              <lb/>
              <figure xlink:label="fig-0217-01" xlink:href="fig-0217-01a" number="187">
                <variables xml:id="echoid-variables176" xml:space="preserve">a o u m h z t s n d ſ e q f p</variables>
              </figure>
            igitur in puncto n.</s>
            <s xml:id="echoid-s14739" xml:space="preserve"> Et [per 31 p 1] à
              <lb/>
            pũcto e ducatur æquidiſtãs lineę
              <lb/>
            t h:</s>
            <s xml:id="echoid-s14740" xml:space="preserve"> & ſit q e:</s>
            <s xml:id="echoid-s14741" xml:space="preserve"> & à puncto e ducatur
              <lb/>
            ęquidiſtãs m z:</s>
            <s xml:id="echoid-s14742" xml:space="preserve"> quę ſit e l.</s>
            <s xml:id="echoid-s14743" xml:space="preserve"> Palã [ք
              <lb/>
            lemma ad 37 th.</s>
            <s xml:id="echoid-s14744" xml:space="preserve"> opticorum Eucli
              <lb/>
            dis:</s>
            <s xml:id="echoid-s14745" xml:space="preserve"> uel per 42 th 6 libri σ{υν}αγωγῶ
              <gap/>
              <lb/>
            μαθκματικῶμ Pappi] quòd m z eſt
              <lb/>
            perpendicularis ſuper a e:</s>
            <s xml:id="echoid-s14746" xml:space="preserve"> quo-
              <lb/>
            niam a h eſt perpendicularis ſu-
              <lb/>
            per circulum, per z tranſeuntem,
              <lb/>
            [per 18 d 11] & m z ſuper diame-
              <lb/>
            trum illius circuli [per 18 p 3]
              <lb/>
            quia contingit.</s>
            <s xml:id="echoid-s14747" xml:space="preserve"> Igitur l e eſt per-
              <lb/>
            pendicularis ſuper a e [per 29
              <lb/>
            p 1] & producatur q e ultra e:</s>
            <s xml:id="echoid-s14748" xml:space="preserve"> hęc
              <lb/>
            concurret quidem cum axe:</s>
            <s xml:id="echoid-s14749" xml:space="preserve"> [per
              <lb/>
            lemma Procli ad 29 p 1] concur-
              <lb/>
            rat in d.</s>
            <s xml:id="echoid-s14750" xml:space="preserve"> Fiat autẽ ſuperficies l e,
              <lb/>
            d q ſecans pyramidem:</s>
            <s xml:id="echoid-s14751" xml:space="preserve"> erit quidem ſectio pyramidalis:</s>
            <s xml:id="echoid-s14752" xml:space="preserve"> [per 5 th.</s>
            <s xml:id="echoid-s14753" xml:space="preserve"> 1 con.</s>
            <s xml:id="echoid-s14754" xml:space="preserve"> Apoll.</s>
            <s xml:id="echoid-s14755" xml:space="preserve"> quia l d q planum ob-
              <lb/>
            liquum eſt ad axem.</s>
            <s xml:id="echoid-s14756" xml:space="preserve">] Cum ergo a e ſit perpendicularis ſuper f n, & ſuper q d, & ſuper l e:</s>
            <s xml:id="echoid-s14757" xml:space="preserve"> erit f n in
              <lb/>
            ſuperficie illa ſecante pyramidem [per 5 p 11.</s>
            <s xml:id="echoid-s14758" xml:space="preserve">] Fiat ergo in illa ſuperficie p f æquidiſtans q e:</s>
            <s xml:id="echoid-s14759" xml:space="preserve"> erit æ-
              <lb/>
            quidiſtans t z [per 9 p 11] uerùm cum angulus f z h ſit acutus:</s>
            <s xml:id="echoid-s14760" xml:space="preserve"> [per concluſionẽ] erit angulus t z
              <gap/>
              <lb/>
            obtuſus [per 13 p 1.</s>
            <s xml:id="echoid-s14761" xml:space="preserve">] Ducatur à puncto z linea, faciens cum t z angulum, æqualem angulo o z t:</s>
            <s xml:id="echoid-s14762" xml:space="preserve"> quę
              <lb/>
            quidem linea neceſſariò ſecabit f p [per lemma Procli ad 29 p 1:</s>
            <s xml:id="echoid-s14763" xml:space="preserve"> quia z t, f p ſunt parallelæ.</s>
            <s xml:id="echoid-s14764" xml:space="preserve">] Secet
              <lb/>
            in puncto p:</s>
            <s xml:id="echoid-s14765" xml:space="preserve"> & ducatur linea p e.</s>
            <s xml:id="echoid-s14766" xml:space="preserve"> Cum ergo p z, o z ſint in eadem ſuperficie, & angulus o z t ęqua-
              <lb/>
            lis angulo t z p [per fabricationem] reflectetur o ad p à puncto ſpeculi z [per 12 n 4.</s>
            <s xml:id="echoid-s14767" xml:space="preserve">] Et quia an-
              <lb/>
            gulus o z t æqualis eſt angulo z f p:</s>
            <s xml:id="echoid-s14768" xml:space="preserve"> [per 29 p 1, & t z p æqualis z p f per eandem:</s>
            <s xml:id="echoid-s14769" xml:space="preserve"> quare z f p,
              <lb/>
            & z p f æquantur] erunt latera z p, z f æqualia [per 6 p 1.</s>
            <s xml:id="echoid-s14770" xml:space="preserve">] Et quia angulus f e z rectus [quia a
              <gap/>
              <lb/>
            perpendicularis eſt ipſ
              <gap/>
            f e n] quadratum f z ualet quadrata e z, e f:</s>
            <s xml:id="echoid-s14771" xml:space="preserve"> & quadratum p z ualet
              <lb/>
            quadrata e z, e p [per 47 p 1.</s>
            <s xml:id="echoid-s14772" xml:space="preserve">] Igitur p e, f e æqualia:</s>
            <s xml:id="echoid-s14773" xml:space="preserve"> [Quia enim z p, z f æquales iam concluſæ
              <lb/>
            ſunt:</s>
            <s xml:id="echoid-s14774" xml:space="preserve"> erunt ipſarum quadrata æqualia:</s>
            <s xml:id="echoid-s14775" xml:space="preserve"> ſubducto igitur communi quadrato z e:</s>
            <s xml:id="echoid-s14776" xml:space="preserve"> relinquentur qua-
              <lb/>
            drata e p, e f æqualia:</s>
            <s xml:id="echoid-s14777" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s14778" xml:space="preserve"> ipſorum latera e p, e f] & ita [per 5 p 1] e p f, e f p anguli erunt
              <lb/>
            æquales.</s>
            <s xml:id="echoid-s14779" xml:space="preserve"> Quare anguli n e q, q e p æquales.</s>
            <s xml:id="echoid-s14780" xml:space="preserve"> [nam per 29 p 1 anguli n e q, e f p:</s>
            <s xml:id="echoid-s14781" xml:space="preserve"> item p e q, e p f
              <lb/>
            æquantur:</s>
            <s xml:id="echoid-s14782" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s14783" xml:space="preserve"> per 1 ax.</s>
            <s xml:id="echoid-s14784" xml:space="preserve"> n e q, p e q æquantur.</s>
            <s xml:id="echoid-s14785" xml:space="preserve">] Et cum in eadem ſuperficie ſint, quæ eſt p en:</s>
            <s xml:id="echoid-s14786" xml:space="preserve"> refle-
              <lb/>
            ctetur n ad p à puncto e [per 12 n 4.</s>
            <s xml:id="echoid-s14787" xml:space="preserve">] Similiter ſi ducatur quæcunq;</s>
            <s xml:id="echoid-s14788" xml:space="preserve"> linea à puncto f a d aliquod pũ-
              <lb/>
            ctum z e, & producatur uſque ad o n:</s>
            <s xml:id="echoid-s14789" xml:space="preserve"> probabitur de puncto lineæ o n, in quod cadit, quòd refle-
              <lb/>
            ctetur ad p à puncto lineæ z e, quod ſecat illa linea.</s>
            <s xml:id="echoid-s14790" xml:space="preserve"> Simili modo & omnium huiuſmodi linea-
              <lb/>
            rum probatio ſumet initium à perpendiculari, quæ eſt f e, & à parte lineæ e z:</s>
            <s xml:id="echoid-s14791" xml:space="preserve"> quæ erit com-
              <lb/>
            munis omnibus illis triangulis.</s>
            <s xml:id="echoid-s14792" xml:space="preserve"> Et ita quo dlibet punctum lineę o n reflectetur ad p ab aliquo pun-
              <lb/>
            cto lineæ e z.</s>
            <s xml:id="echoid-s14793" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div504" type="section" level="0" n="0">
          <head xml:id="echoid-head447" xml:space="preserve" style="it">32. Si linea recta obliquè inciderit uertici ſpeculi conici conuexi: reflectetur à latere coni-
            <lb/>
          co ad uiſum inter dictam lineam & ſpeculi ſuperficiem ſitum: eius́ imago parum curua ui-
            <lb/>
          debitur. 55 p 7.</head>
          <p>
            <s xml:id="echoid-s14794" xml:space="preserve">HOc declarato dicamus.</s>
            <s xml:id="echoid-s14795" xml:space="preserve"> Cum uiſus comprehenderit lineas rectas, tranſeuntes per uerti-
              <lb/>
            cem ſpeculi pyramidalis conuexi recti, obliquas ſuper axem ſpeculi:</s>
            <s xml:id="echoid-s14796" xml:space="preserve"> tunc formæ earum e-
              <lb/>
            runt parùm conuexæ.</s>
            <s xml:id="echoid-s14797" xml:space="preserve"> Sit ergo ſpeculum pyramidale erectum a b c:</s>
            <s xml:id="echoid-s14798" xml:space="preserve"> cuius uertex ſit a:</s>
            <s xml:id="echoid-s14799" xml:space="preserve">
              <lb/>
            & cuius axis ſit a d:</s>
            <s xml:id="echoid-s14800" xml:space="preserve"> & extrahamus in ſuperficie eius lineam a z [ut oſtenſum eſt 52 n 5] quocun-
              <lb/>
            que modo ſit:</s>
            <s xml:id="echoid-s14801" xml:space="preserve"> in qua ſignetur punctum z, quocunque modo ſit.</s>
            <s xml:id="echoid-s14802" xml:space="preserve"> Et tranfeat per z ſuperficies æ-
              <lb/>
            quidiſtans baſi pyramidis:</s>
            <s xml:id="echoid-s14803" xml:space="preserve"> & faciat circulum z u [faciet autem per 4 th 1 con.</s>
            <s xml:id="echoid-s14804" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s14805" xml:space="preserve">] Et extraha-
              <lb/>
            mus ex z perpendicularem z h ſuper a z [per 11 p 1.</s>
            <s xml:id="echoid-s14806" xml:space="preserve">] Hæc ergo linea concurret cum axe pyrami-
              <lb/>
            dis [per 11 ax.</s>
            <s xml:id="echoid-s14807" xml:space="preserve"> ut patuit præcedente numero.</s>
            <s xml:id="echoid-s14808" xml:space="preserve">] Concurrat ergo in h.</s>
            <s xml:id="echoid-s14809" xml:space="preserve"> Et extrahamus ex z line-
              <lb/>
            am contingentem circulum:</s>
            <s xml:id="echoid-s14810" xml:space="preserve"> [per 17 p 3] & ſit z m:</s>
            <s xml:id="echoid-s14811" xml:space="preserve"> & extrahamus ex a lineam continentem cum
              <lb/>
            utraque linea a z, h a angulum acutum:</s>
            <s xml:id="echoid-s14812" xml:space="preserve"> & ſit extra ſuperficiem, contingentem pyramidem, tran-
              <lb/>
            ſeuntem per lineam a z.</s>
            <s xml:id="echoid-s14813" xml:space="preserve"> Et hoc eſt poſsibile:</s>
            <s xml:id="echoid-s14814" xml:space="preserve"> [quia angulus h a z eſt acutus per 18 d 11.</s>
            <s xml:id="echoid-s14815" xml:space="preserve"> 32 p 1.</s>
            <s xml:id="echoid-s14816" xml:space="preserve">] Sit
              <lb/>
            ergo a n:</s>
            <s xml:id="echoid-s14817" xml:space="preserve"> & extrahamus ex puncto h lineam in ſuperficie, in qua ſunt a n, a h, continentem cum
              <lb/>
            a h angulum æqualem angulo a h z.</s>
            <s xml:id="echoid-s14818" xml:space="preserve"> Hæc ergo linea concurret cum
              <gap/>
            o:</s>
            <s xml:id="echoid-s14819" xml:space="preserve"> [per 11 ax.</s>
            <s xml:id="echoid-s14820" xml:space="preserve">] nam
              <lb/>
            duo anguli ad a, h ſunt acuti.</s>
            <s xml:id="echoid-s14821" xml:space="preserve"> Concurrant ergo in o.</s>
            <s xml:id="echoid-s14822" xml:space="preserve"> Linea ergo h o concurret cum cir-
              <lb/>
            cumferentiã circuli z u.</s>
            <s xml:id="echoid-s14823" xml:space="preserve"> Nam angulus a h o eſt æqualis angulo a h z.</s>
            <s xml:id="echoid-s14824" xml:space="preserve"> Concurrat ergo in
              <lb/>
            u:</s>
            <s xml:id="echoid-s14825" xml:space="preserve"> & extrahamus a u rectè:</s>
            <s xml:id="echoid-s14826" xml:space="preserve"> & extrahamus perpendicularem h z ad t:</s>
            <s xml:id="echoid-s14827" xml:space="preserve"> & continuemus o z,
              <lb/>
            & extrahamus rectè ad f:</s>
            <s xml:id="echoid-s14828" xml:space="preserve"> & extrahatur a z ad e.</s>
            <s xml:id="echoid-s14829" xml:space="preserve"> Angulus igitur f z h erit acutus:</s>
            <s xml:id="echoid-s14830" xml:space="preserve"> quia
              <lb/>
            </s>
          </p>
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